Theorem alephnbtwn 4886

Description: No cardinal can be sandwiched between an aleph and its successor aleph. Theorem 67 of [Suppes] p. 229.

Assertion

Ref	Expression
alephnbtwn	|- ((card` B) = B -> -. ((aleph` A) e. B /\ B e. (aleph` suc A)))

Proof of Theorem alephnbtwn

Step	Hyp	Ref	Expression
1		eleq2 1538	. . . . . . 7 |- ((card` B) = B -> ((aleph` A) e. (card` B) <-> (aleph` A) e. B))
2		alephon 4883	. . . . . . . 8 |- (aleph` A) e. On
3		cardsdomel 4870	. . . . . . . 8 |- ((aleph` A) e. On -> ((aleph` A) ~< B <-> (aleph` A) e. (card` B)))
4	2, 3	ax-mp 7	. . . . . . 7 |- ((aleph` A) ~< B <-> (aleph` A) e. (card` B))
5	1, 4	syl5bb 534	. . . . . 6 |- ((card` B) = B -> ((aleph` A) ~< B <-> (aleph` A) e. B))
6	5	adantl 390	. . . . 5 |- ((A e. On /\ (card` B) = B) -> ((aleph` A) ~< B <-> (aleph` A) e. B))
7		alephsuc 4884	. . . . . . . . . 10 |- (A e. On -> (aleph` suc A) = |^|{x e. On | (aleph` A) ~< x})
8	7	eleq2d 1544	. . . . . . . . 9 |- (A e. On -> (B e. (aleph` suc A) <-> B e. |^|{x e. On | (aleph` A) ~< x}))
9	8	biimpd 153	. . . . . . . 8 |- (A e. On -> (B e. (aleph` suc A) -> B e. |^|{x e. On | (aleph` A) ~< x}))
10		breq2 2629	. . . . . . . . 9 |- (x = B -> ((aleph` A) ~< x <-> (aleph` A) ~< B))
11	10	onnminsb 3023	. . . . . . . 8 |- (B e. On -> (B e. |^|{x e. On | (aleph` A) ~< x} -> -. (aleph` A) ~< B))
12	9, 11	sylan9 470	. . . . . . 7 |- ((A e. On /\ B e. On) -> (B e. (aleph` suc A) -> -. (aleph` A) ~< B))
13	12	con2d 91	. . . . . 6 |- ((A e. On /\ B e. On) -> ((aleph` A) ~< B -> -. B e. (aleph` suc A)))
14		cardon 4844	. . . . . . 7 |- (card` B) e. On
15		eleq1 1537	. . . . . . 7 |- ((card` B) = B -> ((card` B) e. On <-> B e. On))
16	14, 15	mpbii 193	. . . . . 6 |- ((card` B) = B -> B e. On)
17	13, 16	sylan2 453	. . . . 5 |- ((A e. On /\ (card` B) = B) -> ((aleph` A) ~< B -> -. B e. (aleph` suc A)))
18	6, 17	sylbird 205	. . . 4 |- ((A e. On /\ (card` B) = B) -> ((aleph` A) e. B -> -. B e. (aleph` suc A)))
19		imnan 242	. . . 4 |- (((aleph` A) e. B -> -. B e. (aleph` suc A)) <-> -. ((aleph` A) e. B /\ B e. (aleph` suc A)))
20	18, 19	sylib 198	. . 3 |- ((A e. On /\ (card` B) = B) -> -. ((aleph` A) e. B /\ B e. (aleph` suc A)))
21	20	ex 373	. 2 |- (A e. On -> ((card` B) = B -> -. ((aleph` A) e. B /\ B e. (aleph` suc A))))
22		n0i 2289	. . . . . . 7 |- (B e. (aleph` suc A) -> -. (aleph` suc A) = (/))
23		alephfnon 4880	. . . . . . . . . . 11 |- aleph Fn On
24		fndm 3594	. . . . . . . . . . 11 |- (aleph Fn On -> dom aleph = On)
25	23, 24	ax-mp 7	. . . . . . . . . 10 |- dom aleph = On
26	25	eleq2i 1541	. . . . . . . . 9 |- (suc A e. dom aleph <-> suc A e. On)
27	26	negbii 187	. . . . . . . 8 |- (-. suc A e. dom aleph <-> -. suc A e. On)
28		ndmfv 3752	. . . . . . . 8 |- (-. suc A e. dom aleph -> (aleph` suc A) = (/))
29	27, 28	sylbir 201	. . . . . . 7 |- (-. suc A e. On -> (aleph` suc A) = (/))
30	22, 29	nsyl2 118	. . . . . 6 |- (B e. (aleph` suc A) -> suc A e. On)
31		sucelon 3075	. . . . . 6 |- (A e. On <-> suc A e. On)
32	30, 31	sylibr 200	. . . . 5 |- (B e. (aleph` suc A) -> A e. On)
33	32	adantl 390	. . . 4 |- (((aleph` A) e. B /\ B e. (aleph` suc A)) -> A e. On)
34	33	con3i 98	. . 3 |- (-. A e. On -> -. ((aleph` A) e. B /\ B e. (aleph` suc A)))
35	34	a1d 12	. 2 |- (-. A e. On -> ((card` B) = B -> -. ((aleph` A) e. B /\ B e. (aleph` suc A))))
36	21, 35	pm2.61i 126	1 |- ((card` B) = B -> -. ((aleph` A) e. B /\ B e. (aleph` suc A)))

Syntax hints:  -. wn 2   -> wi 3   <-> wb 146   /\ wa 223   = wceq 958   e. wcel 960  {crab 1651  (/)c0 2284  |^|cint 2538   class class class wbr 2625  Oncon0 2955  suc csuc 2957  dom cdm 3177   Fn wfn 3184  ` cfv 3189   ~< csdm 4373  cardccrd 4830  alephcale 4831

This theorem is referenced by:  alephnbtwn2 4887

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-9 967  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-rep 2699  ax-sep 2709  ax-nul 2716  ax-pow 2749  ax-pr 2786  ax-un 2873  ax-inf2 4641  ax-ac 4761

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 778  df-3an 779  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-ral 1652  df-rex 1653  df-reu 1654  df-rab 1655  df-v 1815  df-sbc 1945  df-dif 2053  df-un 2054  df-in 2055  df-ss 2057  df-nul 2285  df-if 2367  df-pw 2407  df-sn 2417  df-pr 2418  df-tp 2420  df-op 2421  df-uni 2509  df-int 2539  df-iun 2573  df-br 2626  df-opab 2673  df-tr 2687  df-eprel 2839  df-id 2842  df-po 2847  df-so 2857  df-fr 2924  df-we 2941  df-ord 2958  df-on 2959  df-lim 2960  df-suc 2961  df-om 3139  df-xp 3191  df-rel 3192  df-cnv 3193  df-co 3194  df-dm 3195  df-rn 3196  df-res 3197  df-ima 3198  df-fun 3199  df-fn 3200  df-f 3201  df-f1 3202  df-fo 3203  df-f1o 3204  df-fv 3205  df-rdg 3939  df-er 4268  df-en 4375  df-dom 4376  df-sdom 4377  df-card 4833  df-aleph 4834

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