Theorem alephinit 8476

Description: An infinite initial ordinal is characterized by the property of being initial - that is, it is a subset of any dominating ordinal. (Contributed by Jeff Hankins, 29-Oct-2009.) (Proof shortened by Mario Carneiro, 20-Sep-2014.)

Assertion

Ref	Expression
alephinit	|- ( ( A e. On /\ om C_ A ) -> ( A e. ran aleph <-> A. x e. On ( A ~<_ x -> A C_ x ) ) )

Distinct variable group:   x, A

Proof of Theorem alephinit

Step	Hyp	Ref	Expression
1		isinfcard 8473	. . . . 5 |- ( ( om C_ A /\ ( card ` A ) = A ) <-> A e. ran aleph )
2	1	bicomi 202	. . . 4 |- ( A e. ran aleph <-> ( om C_ A /\ ( card ` A ) = A ) )
3	2	baib 901	. . 3 |- ( om C_ A -> ( A e. ran aleph <-> ( card ` A ) = A ) )
4	3	adantl 466	. 2 |- ( ( A e. On /\ om C_ A ) -> ( A e. ran aleph <-> ( card ` A ) = A ) )
5		onenon 8330	. . . . . . . 8 |- ( A e. On -> A e. dom card )
6	5	adantr 465	. . . . . . 7 |- ( ( A e. On /\ om C_ A ) -> A e. dom card )
7		onenon 8330	. . . . . . 7 |- ( x e. On -> x e. dom card )
8		carddom2 8358	. . . . . . 7 |- ( ( A e. dom card /\ x e. dom card ) -> ( ( card ` A ) C_ ( card ` x ) <-> A ~<_ x ) )
9	6, 7, 8	syl2an 477	. . . . . 6 |- ( ( ( A e. On /\ om C_ A ) /\ x e. On ) -> ( ( card ` A ) C_ ( card ` x ) <-> A ~<_ x ) )
10		cardonle 8338	. . . . . . . 8 |- ( x e. On -> ( card ` x ) C_ x )
11	10	adantl 466	. . . . . . 7 |- ( ( ( A e. On /\ om C_ A ) /\ x e. On ) -> ( card ` x ) C_ x )
12		sstr 3512	. . . . . . . 8 |- ( ( ( card ` A ) C_ ( card ` x ) /\ ( card ` x ) C_ x ) -> ( card ` A ) C_ x )
13	12	expcom 435	. . . . . . 7 |- ( ( card ` x ) C_ x -> ( ( card ` A ) C_ ( card ` x ) -> ( card ` A ) C_ x ) )
14	11, 13	syl 16	. . . . . 6 |- ( ( ( A e. On /\ om C_ A ) /\ x e. On ) -> ( ( card ` A ) C_ ( card ` x ) -> ( card ` A ) C_ x ) )
15	9, 14	sylbird 235	. . . . 5 |- ( ( ( A e. On /\ om C_ A ) /\ x e. On ) -> ( A ~<_ x -> ( card ` A ) C_ x ) )
16		sseq1 3525	. . . . . 6 |- ( ( card ` A ) = A -> ( ( card ` A ) C_ x <-> A C_ x ) )
17	16	imbi2d 316	. . . . 5 |- ( ( card ` A ) = A -> ( ( A ~<_ x -> ( card ` A ) C_ x ) <-> ( A ~<_ x -> A C_ x ) ) )
18	15, 17	syl5ibcom 220	. . . 4 |- ( ( ( A e. On /\ om C_ A ) /\ x e. On ) -> ( ( card ` A ) = A -> ( A ~<_ x -> A C_ x ) ) )
19	18	ralrimdva 2882	. . 3 |- ( ( A e. On /\ om C_ A ) -> ( ( card ` A ) = A -> A. x e. On ( A ~<_ x -> A C_ x ) ) )
20		oncardid 8337	. . . . . . 7 |- ( A e. On -> ( card ` A ) ~~ A )
21		ensym 7564	. . . . . . 7 |- ( ( card ` A ) ~~ A -> A ~~ ( card ` A ) )
22		endom 7542	. . . . . . 7 |- ( A ~~ ( card ` A ) -> A ~<_ ( card ` A ) )
23	20, 21, 22	3syl 20	. . . . . 6 |- ( A e. On -> A ~<_ ( card ` A ) )
24	23	adantr 465	. . . . 5 |- ( ( A e. On /\ om C_ A ) -> A ~<_ ( card ` A ) )
25		cardon 8325	. . . . . 6 |- ( card ` A ) e. On
26		breq2 4451	. . . . . . . 8 |- ( x = ( card ` A ) -> ( A ~<_ x <-> A ~<_ ( card ` A ) ) )
27		sseq2 3526	. . . . . . . 8 |- ( x = ( card ` A ) -> ( A C_ x <-> A C_ ( card ` A ) ) )
28	26, 27	imbi12d 320	. . . . . . 7 |- ( x = ( card ` A ) -> ( ( A ~<_ x -> A C_ x ) <-> ( A ~<_ ( card ` A ) -> A C_ ( card ` A ) ) ) )
29	28	rspcv 3210	. . . . . 6 |- ( ( card ` A ) e. On -> ( A. x e. On ( A ~<_ x -> A C_ x ) -> ( A ~<_ ( card ` A ) -> A C_ ( card ` A ) ) ) )
30	25, 29	ax-mp 5	. . . . 5 |- ( A. x e. On ( A ~<_ x -> A C_ x ) -> ( A ~<_ ( card ` A ) -> A C_ ( card ` A ) ) )
31	24, 30	syl5com 30	. . . 4 |- ( ( A e. On /\ om C_ A ) -> ( A. x e. On ( A ~<_ x -> A C_ x ) -> A C_ ( card ` A ) ) )
32		cardonle 8338	. . . . . . 7 |- ( A e. On -> ( card ` A ) C_ A )
33	32	adantr 465	. . . . . 6 |- ( ( A e. On /\ om C_ A ) -> ( card ` A ) C_ A )
34	33	biantrurd 508	. . . . 5 |- ( ( A e. On /\ om C_ A ) -> ( A C_ ( card ` A ) <-> ( ( card ` A ) C_ A /\ A C_ ( card ` A ) ) ) )
35		eqss 3519	. . . . 5 |- ( ( card ` A ) = A <-> ( ( card ` A ) C_ A /\ A C_ ( card ` A ) ) )
36	34, 35	syl6bbr 263	. . . 4 |- ( ( A e. On /\ om C_ A ) -> ( A C_ ( card ` A ) <-> ( card ` A ) = A ) )
37	31, 36	sylibd 214	. . 3 |- ( ( A e. On /\ om C_ A ) -> ( A. x e. On ( A ~<_ x -> A C_ x ) -> ( card ` A ) = A ) )
38	19, 37	impbid 191	. 2 |- ( ( A e. On /\ om C_ A ) -> ( ( card ` A ) = A <-> A. x e. On ( A ~<_ x -> A C_ x ) ) )
39	4, 38	bitrd 253	1 |- ( ( A e. On /\ om C_ A ) -> ( A e. ran aleph <-> A. x e. On ( A ~<_ x -> A C_ x ) ) )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 369   = wceq 1379   e. wcel 1767   A.wral 2814   C_ wss 3476   class class class wbr 4447   Oncon0 4878   dom cdm 4999   ran crn 5000   ` cfv 5588   omcom 6684   ~~ cen 7513   ~<_ cdom 7514   cardccrd 8316   alephcale 8317

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576  ax-inf2 8058

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-se 4839  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-isom 5597  df-riota 6245  df-om 6685  df-recs 7042  df-rdg 7076  df-er 7311  df-en 7517  df-dom 7518  df-sdom 7519  df-fin 7520  df-oi 7935  df-har 7984  df-card 8320  df-aleph 8321

This theorem is referenced by: (None)