Theorem php 4513

Description: Pigeonhole Principle. A natural number is not equinumerous to a proper subset of itself. Theorem (Pigeonhole Principle) of [Enderton] p. 134. The theorem is so-called because you can't put n + 1 pigeons into n holes (if each hole holds only one pigeon). The proof consists of lemmas phplem1 4508 through phplem4 4511, nneneq 4512, and this final piece of the proof.

Assertion

Ref	Expression
php	|- ((A e. om /\ B (. A) -> -. A ~~ B)

Proof of Theorem php

Step	Hyp	Ref	Expression
1		nn0suc 3154	. . . . . . 7 |- (A e. om -> (A = (/) \/ E.x e. om A = suc x))
2	1	orcanai 690	. . . . . 6 |- ((A e. om /\ -. A = (/)) -> E.x e. om A = suc x)
3		0ss 2301	. . . . . . . 8 |- (/) (_ B
4		sspsstr 2151	. . . . . . . 8 |- (((/) (_ B /\ B (. A) -> (/) (. A)
5	3, 4	mpan 695	. . . . . . 7 |- (B (. A -> (/) (. A)
6		0pss 2308	. . . . . . . 8 |- ((/) (. A <-> A =/= (/))
7		df-ne 1587	. . . . . . . 8 |- (A =/= (/) <-> -. A = (/))
8	6, 7	bitr 173	. . . . . . 7 |- ((/) (. A <-> -. A = (/))
9	5, 8	sylib 198	. . . . . 6 |- (B (. A -> -. A = (/))
10	2, 9	sylan2 451	. . . . 5 |- ((A e. om /\ B (. A) -> E.x e. om A = suc x)
11		psseq2 2136	. . . . . . . 8 |- (A = suc x -> (B (. A <-> B (. suc x))
12		breq1 2622	. . . . . . . . 9 |- (A = suc x -> (A ~~ B <-> suc x ~~ B))
13	12	negbid 611	. . . . . . . 8 |- (A = suc x -> (-. A ~~ B <-> -. suc x ~~ B))
14	11, 13	imbi12d 626	. . . . . . 7 |- (A = suc x -> ((B (. A -> -. A ~~ B) <-> (B (. suc x -> -. suc x ~~ B)))
15		pssnel 2331	. . . . . . . . . 10 |- (B (. suc x -> E.y(y e. suc x /\ -. y e. B))
16		domentr 4421	. . . . . . . . . . . . . . 15 |- ((B ~<_ (suc x \ {y}) /\ (suc x \ {y}) ~~ x) -> B ~<_ x)
17		disjsn 2441	. . . . . . . . . . . . . . . . . . . . 21 |- ((B i^i {y}) = (/) <-> -. y e. B)
18		disj3 2314	. . . . . . . . . . . . . . . . . . . . 21 |- ((B i^i {y}) = (/) <-> B = (B \ {y}))
19	17, 18	bitr3 175	. . . . . . . . . . . . . . . . . . . 20 |- (-. y e. B <-> B = (B \ {y}))
20		sseq1 2082	. . . . . . . . . . . . . . . . . . . 20 |- (B = (B \ {y}) -> (B (_ (suc x \ {y}) <-> (B \ {y}) (_ (suc x \ {y})))
21	19, 20	sylbi 199	. . . . . . . . . . . . . . . . . . 19 |- (-. y e. B -> (B (_ (suc x \ {y}) <-> (B \ {y}) (_ (suc x \ {y})))
22		ssdif 2172	. . . . . . . . . . . . . . . . . . 19 |- (B (_ suc x -> (B \ {y}) (_ (suc x \ {y}))
23	21, 22	syl5bir 210	. . . . . . . . . . . . . . . . . 18 |- (-. y e. B -> (B (_ suc x -> B (_ (suc x \ {y})))
24		visset 1813	. . . . . . . . . . . . . . . . . . . . 21 |- x e. V
25	24	sucex 3050	. . . . . . . . . . . . . . . . . . . 20 |- suc x e. V
26		difss 2167	. . . . . . . . . . . . . . . . . . . 20 |- (suc x \ {y}) (_ suc x
27	25, 26	ssexi 2720	. . . . . . . . . . . . . . . . . . 19 |- (suc x \ {y}) e. V
28		ssdom2g 4409	. . . . . . . . . . . . . . . . . . 19 |- ((suc x \ {y}) e. V -> (B (_ (suc x \ {y}) -> B ~<_ (suc x \ {y})))
29	27, 28	ax-mp 7	. . . . . . . . . . . . . . . . . 18 |- (B (_ (suc x \ {y}) -> B ~<_ (suc x \ {y}))
30	23, 29	syl6 22	. . . . . . . . . . . . . . . . 17 |- (-. y e. B -> (B (_ suc x -> B ~<_ (suc x \ {y})))
31		pssss 2143	. . . . . . . . . . . . . . . . 17 |- (B (. suc x -> B (_ suc x)
32	30, 31	syl5 21	. . . . . . . . . . . . . . . 16 |- (-. y e. B -> (B (. suc x -> B ~<_ (suc x \ {y})))
33	32	imp 350	. . . . . . . . . . . . . . 15 |- ((-. y e. B /\ B (. suc x) -> B ~<_ (suc x \ {y}))
34		visset 1813	. . . . . . . . . . . . . . . . 17 |- y e. V
35	24, 34	phplem3 4510	. . . . . . . . . . . . . . . 16 |- ((x e. om /\ y e. suc x) -> x ~~ (suc x \ {y}))
36	27	ensym 4412	. . . . . . . . . . . . . . . 16 |- (x ~~ (suc x \ {y}) -> (suc x \ {y}) ~~ x)
37	35, 36	syl 10	. . . . . . . . . . . . . . 15 |- ((x e. om /\ y e. suc x) -> (suc x \ {y}) ~~ x)
38	16, 33, 37	syl2an 454	. . . . . . . . . . . . . 14 |- (((-. y e. B /\ B (. suc x) /\ (x e. om /\ y e. suc x)) -> B ~<_ x)
39	38	exp43 384	. . . . . . . . . . . . 13 |- (-. y e. B -> (B (. suc x -> (x e. om -> (y e. suc x -> B ~<_ x))))
40	39	com4r 41	. . . . . . . . . . . 12 |- (y e. suc x -> (-. y e. B -> (B (. suc x -> (x e. om -> B ~<_ x))))
41	40	imp 350	. . . . . . . . . . 11 |- ((y e. suc x /\ -. y e. B) -> (B (. suc x -> (x e. om -> B ~<_ x)))
42	41	19.23aiv 1295	. . . . . . . . . 10 |- (E.y(y e. suc x /\ -. y e. B) -> (B (. suc x -> (x e. om -> B ~<_ x)))
43	15, 42	mpcom 49	. . . . . . . . 9 |- (B (. suc x -> (x e. om -> B ~<_ x))
44		endomtr 4420	. . . . . . . . . . . . 13 |- ((suc x ~~ B /\ B ~<_ x) -> suc x ~<_ x)
45		sssucid 3047	. . . . . . . . . . . . . 14 |- x (_ suc x
46		ssdom2g 4409	. . . . . . . . . . . . . 14 |- (suc x e. V -> (x (_ suc x -> x ~<_ suc x))
47	25, 45, 46	mp2 43	. . . . . . . . . . . . 13 |- x ~<_ suc x
48	44, 47	jctir 293	. . . . . . . . . . . 12 |- ((suc x ~~ B /\ B ~<_ x) -> (suc x ~<_ x /\ x ~<_ suc x))
49		sbth 4457	. . . . . . . . . . . 12 |- ((suc x ~<_ x /\ x ~<_ suc x) -> suc x ~~ x)
50	48, 49	syl 10	. . . . . . . . . . 11 |- ((suc x ~~ B /\ B ~<_ x) -> suc x ~~ x)
51	50	expcom 374	. . . . . . . . . 10 |- (B ~<_ x -> (suc x ~~ B -> suc x ~~ x))
52		peano2b 3147	. . . . . . . . . . . . 13 |- (x e. om <-> suc x e. om)
53		nnord 3140	. . . . . . . . . . . . 13 |- (suc x e. om -> Ord suc x)
54	52, 53	sylbi 199	. . . . . . . . . . . 12 |- (x e. om -> Ord suc x)
55	24	sucid 3051	. . . . . . . . . . . . 13 |- x e. suc x
56		nordeq 2967	. . . . . . . . . . . . 13 |- ((Ord suc x /\ x e. suc x) -> suc x =/= x)
57	55, 56	mpan2 696	. . . . . . . . . . . 12 |- (Ord suc x -> suc x =/= x)
58	54, 57	syl 10	. . . . . . . . . . 11 |- (x e. om -> suc x =/= x)
59		nneneq 4512	. . . . . . . . . . . . . 14 |- ((suc x e. om /\ x e. om) -> (suc x ~~ x <-> suc x = x))
60	59, 52	sylanb 449	. . . . . . . . . . . . 13 |- ((x e. om /\ x e. om) -> (suc x ~~ x <-> suc x = x))
61	60	anidms 434	. . . . . . . . . . . 12 |- (x e. om -> (suc x ~~ x <-> suc x = x))
62	61	necon3bbid 1600	. . . . . . . . . . 11 |- (x e. om -> (-. suc x ~~ x <-> suc x =/= x))
63	58, 62	mpbird 196	. . . . . . . . . 10 |- (x e. om -> -. suc x ~~ x)
64	51, 63	nsyli 121	. . . . . . . . 9 |- (B ~<_ x -> (x e. om -> -. suc x ~~ B))
65	43, 64	syli 54	. . . . . . . 8 |- (B (. suc x -> (x e. om -> -. suc x ~~ B))
66	65	com12 11	. . . . . . 7 |- (x e. om -> (B (. suc x -> -. suc x ~~ B))
67	14, 66	syl5cbir 211	. . . . . 6 |- (x e. om -> (A = suc x -> (B (. A -> -. A ~~ B)))
68	67	r19.23aiv 1743	. . . . 5 |- (E.x e. om A = suc x -> (B (. A -> -. A ~~ B))
69	10, 68	syl 10	. . . 4 |- ((A e. om /\ B (. A) -> (B (. A -> -. A ~~ B))
70	69	ex 373	. . 3 |- (A e. om -> (B (. A -> (B (. A -> -. A ~~ B)))
71	70	pm2.43d 65	. 2 |- (A e. om -> (B (. A -> -. A ~~ B))
72	71	imp 350	1 |- ((A e. om /\ B (. A) -> -. A ~~ B)

Syntax hints:  -. wn 2   -> wi 3   <-> wb 146   /\ wa 223   = wceq 956   e. wcel 958  E.wex 980   =/= wne 1585  E.wrex 1646  Vcvv 1811   \ cdif 2044   i^i cin 2046   (_ wss 2047   (. wpss 2048  (/)c0 2280  {csn 2409   class class class wbr 2619  Ord word 2947  suc csuc 2950  omcom 3131   ~~ cen 4364   ~<_ cdom 4365

This theorem is referenced by:  php2 4514  php3 4515  php3OLD 4516  top2usne 10549

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-9 965  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-rep 2693  ax-sep 2703  ax-nul 2710  ax-pow 2742  ax-pr 2779  ax-un 2866

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 776  df-3an 777  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-ral 1649  df-rex 1650  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-pss 2055  df-nul 2281  df-if 2362  df-pw 2402  df-sn 2412  df-pr 2413  df-tp 2415  df-op 2416  df-uni 2504  df-br 2620  df-opab 2667  df-tr 2681  df-eprel 2832  df-id 2835  df-po 2840  df-so 2850  df-fr 2917  df-we 2934  df-ord 2951  df-on 2952  df-lim 2953  df-suc 2954  df-om 3132  df-xp 3184  df-rel 3185  df-cnv 3186  df-co 3187  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191  df-fun 3192  df-fn 3193  df-f 3194  df-f1 3195  df-fo 3196  df-f1o 3197  df-fv 3198  df-er 4261  df-en 4368  df-dom 4369

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