Theorem isinfcard 6035

Description: Two ways to express the property of being a transfinite cardinal.

Assertion

Ref	Expression
isinfcard	|- ((om C_ A /\ (card` A) = A) <-> A e. ran aleph)

Proof of Theorem isinfcard

Step	Hyp	Ref	Expression
1		eqcom 1886	. . 3 |- ((aleph` x) = A <-> A = (aleph` x))
2	1	rexbii 2128	. 2 |- (E.x e. On (aleph` x) = A <-> E.x e. On A = (aleph` x))
3		alephfnon 5873	. . 3 |- aleph Fn On
4		fvelrnb 4719	. . 3 |- (aleph Fn On -> (A e. ran aleph <-> E.x e. On (aleph` x) = A))
5	3, 4	ax-mp 7	. 2 |- (A e. ran aleph <-> E.x e. On (aleph` x) = A)
6		cardalephex 6034	. . . 4 |- (om C_ A -> ((card` A) = A <-> E.x e. On A = (aleph` x)))
7	6	pm5.32i 707	. . 3 |- ((om C_ A /\ (card` A) = A) <-> (om C_ A /\ E.x e. On A = (aleph` x)))
8		sseq2 2639	. . . . . 6 |- (A = (aleph` x) -> (om C_ A <-> om C_ (aleph` x)))
9		alephgeom 6030	. . . . . . 7 |- (x e. On <-> om C_ (aleph` x))
10	9	biimpi 168	. . . . . 6 |- (x e. On -> om C_ (aleph` x))
11	8, 10	syl5cbir 228	. . . . 5 |- (x e. On -> (A = (aleph` x) -> om C_ A))
12	11	r19.23aiv 2211	. . . 4 |- (E.x e. On A = (aleph` x) -> om C_ A)
13	12	pm4.71ri 700	. . 3 |- (E.x e. On A = (aleph` x) <-> (om C_ A /\ E.x e. On A = (aleph` x)))
14	7, 13	bitr4i 193	. 2 |- ((om C_ A /\ (card` A) = A) <-> E.x e. On A = (aleph` x))
15	2, 5, 14	3bitr4ri 201	1 |- ((om C_ A /\ (card` A) = A) <-> A e. ran aleph)

Syntax hints:   <-> wb 163   /\ wa 240   = wceq 1298   e. wcel 1300  E.wrex 2106   C_ wss 2593  Oncon0 3657  omcom 3949  ran crn 3987   Fn wfn 3993  ` cfv 3998  cardccrd 5859  alephcale 5860

This theorem is referenced by:  iscard3 6036  cardinfima 6039  alephiso 6040  alephsson 6042  alephfp 6048

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790  ax-reg 5695  ax-inf2 5731  ax-ac 5906

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3or 859  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-reu 2111  df-rab 2112  df-v 2294  df-sbc 2454  df-csb 2541  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-pss 2607  df-nul 2876  df-if 2983  df-pw 3035  df-sn 3049  df-pr 3050  df-tp 3052  df-op 3053  df-uni 3178  df-int 3215  df-iun 3257  df-br 3339  df-opab 3396  df-tr 3412  df-eprel 3583  df-id 3586  df-po 3591  df-so 3604  df-fr 3625  df-we 3644  df-ord 3660  df-on 3661  df-lim 3662  df-suc 3663  df-om 3950  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-f 4010  df-f1 4011  df-fo 4012  df-f1o 4013  df-fv 4014  df-rdg 5140  df-er 5318  df-en 5427  df-dom 5428  df-sdom 5429  df-fin 5430  df-card 5862  df-aleph 5863

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