Theorem alephinit 8493

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 8490	. . . . 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 903	. . 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 8347	. . . . . . . 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 8347	. . . . . . 7 |- ( x e. On -> x e. dom card )
8		carddom2 8375	. . . . . . 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 8355	. . . . . . . 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 3507	. . . . . . . 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 3520	. . . . . 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 2875	. . 3 |- ( ( A e. On /\ om C_ A ) -> ( ( card ` A ) = A -> A. x e. On ( A ~<_ x -> A C_ x ) ) )
20		oncardid 8354	. . . . . . 7 |- ( A e. On -> ( card ` A ) ~~ A )
21		ensym 7583	. . . . . . 7 |- ( ( card ` A ) ~~ A -> A ~~ ( card ` A ) )
22		endom 7561	. . . . . . 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 8342	. . . . . 6 |- ( card ` A ) e. On
26		breq2 4460	. . . . . . . 8 |- ( x = ( card ` A ) -> ( A ~<_ x <-> A ~<_ ( card ` A ) ) )
27		sseq2 3521	. . . . . . . 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 3206	. . . . . 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 8355	. . . . . . 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 3514	. . . . 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 1395   e. wcel 1819   A.wral 2807   C_ wss 3471   class class class wbr 4456   Oncon0 4887   dom cdm 5008   ran crn 5009   ` cfv 5594   omcom 6699   ~~ cen 7532   ~<_ cdom 7533   cardccrd 8333   alephcale 8334

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-inf2 8075

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-se 4848  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-isom 5603  df-riota 6258  df-om 6700  df-recs 7060  df-rdg 7094  df-er 7329  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-oi 7953  df-har 8002  df-card 8337  df-aleph 8338

This theorem is referenced by: (None)