Theorem isinfcard 8915

Description: Two ways to express the property of being a transfinite cardinal. (Contributed by NM, 9-Nov-2003.)

Assertion

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

Proof of Theorem isinfcard

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		alephfnon 8888	. . 3 |- aleph Fn On
2		fvelrnb 6243	. . 3 |- ( aleph Fn On -> ( A e. ran aleph <-> E. x e. On ( aleph ` x ) = A ) )
3	1, 2	ax-mp 5	. 2 |- ( A e. ran aleph <-> E. x e. On ( aleph ` x ) = A )
4		alephgeom 8905	. . . . . . 7 |- ( x e. On <-> om C_ ( aleph ` x ) )
5	4	biimpi 206	. . . . . 6 |- ( x e. On -> om C_ ( aleph ` x ) )
6		sseq2 3627	. . . . . 6 |- ( A = ( aleph ` x ) -> ( om C_ A <-> om C_ ( aleph ` x ) ) )
7	5, 6	syl5ibrcom 237	. . . . 5 |- ( x e. On -> ( A = ( aleph ` x ) -> om C_ A ) )
8	7	rexlimiv 3027	. . . 4 |- ( E. x e. On A = ( aleph ` x ) -> om C_ A )
9	8	pm4.71ri 665	. . 3 |- ( E. x e. On A = ( aleph ` x ) <-> ( om C_ A /\ E. x e. On A = ( aleph ` x ) ) )
10		eqcom 2629	. . . 4 |- ( ( aleph ` x ) = A <-> A = ( aleph ` x ) )
11	10	rexbii 3041	. . 3 |- ( E. x e. On ( aleph ` x ) = A <-> E. x e. On A = ( aleph ` x ) )
12		cardalephex 8913	. . . 4 |- ( om C_ A -> ( ( card ` A ) = A <-> E. x e. On A = ( aleph ` x ) ) )
13	12	pm5.32i 669	. . 3 |- ( ( om C_ A /\ ( card ` A ) = A ) <-> ( om C_ A /\ E. x e. On A = ( aleph ` x ) ) )
14	9, 11, 13	3bitr4i 292	. 2 |- ( E. x e. On ( aleph ` x ) = A <-> ( om C_ A /\ ( card ` A ) = A ) )
15	3, 14	bitr2i 265	1 |- ( ( om C_ A /\ ( card ` A ) = A ) <-> A e. ran aleph )

Syntax hints:   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   E.wrex 2913   C_ wss 3574   ran crn 5115   Oncon0 5723   Fn wfn 5883   ` cfv 5888   omcom 7065   cardccrd 8761   alephcale 8762

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-om 7066  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-oi 8415  df-har 8463  df-card 8765  df-aleph 8766

This theorem is referenced by:  iscard3  8916  alephinit  8918  cardinfima  8920  alephiso  8921  alephsson  8923  alephfp  8931