Theorem alephon 4865

Description: An aleph is an ordinal number.

Assertion

Ref	Expression
alephon	|- (aleph` A) e. On

Proof of Theorem alephon

Step	Hyp	Ref	Expression
1		fveq2 3724	. . . 4 |- (x = (/) -> (aleph` x) = (aleph` (/)))
2	1	eleq1d 1540	. . 3 |- (x = (/) -> ((aleph` x) e. On <-> (aleph` (/)) e. On))
3		fveq2 3724	. . . 4 |- (x = y -> (aleph` x) = (aleph` y))
4	3	eleq1d 1540	. . 3 |- (x = y -> ((aleph` x) e. On <-> (aleph` y) e. On))
5		fveq2 3724	. . . 4 |- (x = suc y -> (aleph` x) = (aleph` suc y))
6	5	eleq1d 1540	. . 3 |- (x = suc y -> ((aleph` x) e. On <-> (aleph` suc y) e. On))
7		fveq2 3724	. . . 4 |- (x = A -> (aleph` x) = (aleph` A))
8	7	eleq1d 1540	. . 3 |- (x = A -> ((aleph` x) e. On <-> (aleph` A) e. On))
9		aleph0 4863	. . . 4 |- (aleph` (/)) = om
10		omelon 4629	. . . 4 |- om e. On
11	9, 10	eqeltr 1544	. . 3 |- (aleph` (/)) e. On
12		ax-17 971	. . . . . . . . . 10 |- (w e. om -> A.z w e. om)
13		ax-17 971	. . . . . . . . . 10 |- (w e. y -> A.z w e. y)
14		ax-17 971	. . . . . . . . . 10 |- (w e. |^|{x e. On | (aleph` y) ~< x} -> A.z w e. |^|{x e. On | (aleph` y) ~< x})
15		df-aleph 4817	. . . . . . . . . 10 |- aleph = rec({<.z, y>. | y = |^|{x e. On | z ~< x}}, om)
16		breq1 2622	. . . . . . . . . . . 12 |- (z = (aleph` y) -> (z ~< x <-> (aleph` y) ~< x))
17	16	rabbisdv 1807	. . . . . . . . . . 11 |- (z = (aleph` y) -> {x e. On | z ~< x} = {x e. On | (aleph` y) ~< x})
18	17	inteqd 2538	. . . . . . . . . 10 |- (z = (aleph` y) -> |^|{x e. On | z ~< x} = |^|{x e. On | (aleph` y) ~< x})
19	12, 13, 14, 15, 18	rdgsucopab 3946	. . . . . . . . 9 |- ((y e. On /\ |^|{x e. On | (aleph` y) ~< x} e. V) -> (aleph` suc y) = |^|{x e. On | (aleph` y) ~< x})
20	19	eleq1d 1540	. . . . . . . 8 |- ((y e. On /\ |^|{x e. On | (aleph` y) ~< x} e. V) -> ((aleph` suc y) e. On <-> |^|{x e. On | (aleph` y) ~< x} e. On))
21		onintrab 3013	. . . . . . . 8 |- (|^|{x e. On | (aleph` y) ~< x} e. V <-> |^|{x e. On | (aleph` y) ~< x} e. On)
22	20, 21	syl6rbbr 539	. . . . . . 7 |- ((y e. On /\ |^|{x e. On | (aleph` y) ~< x} e. V) -> (|^|{x e. On | (aleph` y) ~< x} e. V <-> (aleph` suc y) e. On))
23	22	ex 373	. . . . . 6 |- (y e. On -> (|^|{x e. On | (aleph` y) ~< x} e. V -> (|^|{x e. On | (aleph` y) ~< x} e. V <-> (aleph` suc y) e. On)))
24	23	ibd 594	. . . . 5 |- (y e. On -> (|^|{x e. On | (aleph` y) ~< x} e. V -> (aleph` suc y) e. On))
25	12, 13, 14, 15, 18	rdgsucopabn 3947	. . . . . 6 |- (-. |^|{x e. On | (aleph` y) ~< x} e. V -> (aleph` suc y) = (/))
26		0elon 3022	. . . . . 6 |- (/) e. On
27	25, 26	syl6eqel 1556	. . . . 5 |- (-. |^|{x e. On | (aleph` y) ~< x} e. V -> (aleph` suc y) e. On)
28	24, 27	pm2.61d1 128	. . . 4 |- (y e. On -> (aleph` suc y) e. On)
29	28	a1d 12	. . 3 |- (y e. On -> ((aleph` y) e. On -> (aleph` suc y) e. On))
30		visset 1813	. . . . . 6 |- x e. V
31		alephlim 4864	. . . . . 6 |- ((x e. V /\ Lim x) -> (aleph` x) = U_y e. x (aleph` y))
32	30, 31	mpan 695	. . . . 5 |- (Lim x -> (aleph` x) = U_y e. x (aleph` y))
33	32	eleq1d 1540	. . . 4 |- (Lim x -> ((aleph` x) e. On <-> U_y e. x (aleph` y) e. On))
34		fvex 3732	. . . . 5 |- (aleph` y) e. V
35	30, 34	iunon 3909	. . . 4 |- (A.y e. x (aleph` y) e. On -> U_y e. x (aleph` y) e. On)
36	33, 35	syl5bir 210	. . 3 |- (Lim x -> (A.y e. x (aleph` y) e. On -> (aleph` x) e. On))
37	2, 4, 6, 8, 11, 29, 36	tfinds 3161	. 2 |- (A e. On -> (aleph` A) e. On)
38		alephfnon 4862	. . . . . . 7 |- aleph Fn On
39		fndm 3587	. . . . . . 7 |- (aleph Fn On -> dom aleph = On)
40	38, 39	ax-mp 7	. . . . . 6 |- dom aleph = On
41	40	eleq2i 1538	. . . . 5 |- (A e. dom aleph <-> A e. On)
42	41	negbii 187	. . . 4 |- (-. A e. dom aleph <-> -. A e. On)
43		ndmfv 3745	. . . 4 |- (-. A e. dom aleph -> (aleph` A) = (/))
44	42, 43	sylbir 201	. . 3 |- (-. A e. On -> (aleph` A) = (/))
45	44, 26	syl6eqel 1556	. 2 |- (-. A e. On -> (aleph` A) e. On)
46	37, 45	pm2.61i 126	1 |- (aleph` A) e. On

Syntax hints:  -. wn 2   <-> wb 146   /\ wa 223   = wceq 956   e. wcel 958  A.wral 1645  {crab 1648  Vcvv 1811  (/)c0 2280  |^|cint 2533  U_ciun 2566   class class class wbr 2619  Oncon0 2948  Lim wlim 2949  suc csuc 2950  omcom 3131  dom cdm 3170   Fn wfn 3177  ` cfv 3182   ~< csdm 4366  alephcale 4814

This theorem is referenced by:  alephnbtwn 4868  alephnbtwn2 4869  alephordlem1 4872  alephordlem2 4873  alephordi 4874  alephord 4875  alephord2 4876  alephord3 4878  alephle 4884  cardaleph 4885  alephfp 4900  alephval2 4902

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  ax-inf2 4625

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-rab 1652  df-v 1812  df-sbc 1942  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  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-int 2534  df-iun 2568  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-fv 3198  df-rdg 3932  df-aleph 4817

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