Theorem alephinit 8270

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 8267	. . . . 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 896	. . 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 8124	. . . . . . . 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 8124	. . . . . . 7 |- ( x e. On -> x e. dom card )
8		carddom2 8152	. . . . . . 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 8132	. . . . . . . 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 3369	. . . . . . . 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 3382	. . . . . 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 2811	. . 3 |- ( ( A e. On /\ om C_ A ) -> ( ( card ` A ) = A -> A. x e. On ( A ~<_ x -> A C_ x ) ) )
20		oncardid 8131	. . . . . . 7 |- ( A e. On -> ( card ` A ) ~~ A )
21		ensym 7363	. . . . . . 7 |- ( ( card ` A ) ~~ A -> A ~~ ( card ` A ) )
22		endom 7341	. . . . . . 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 8119	. . . . . 6 |- ( card ` A ) e. On
26		breq2 4301	. . . . . . . 8 |- ( x = ( card ` A ) -> ( A ~<_ x <-> A ~<_ ( card ` A ) ) )
27		sseq2 3383	. . . . . . . 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 3074	. . . . . 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 8132	. . . . . . 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 3376	. . . . 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 1369   e. wcel 1756   A.wral 2720   C_ wss 3333   class class class wbr 4297   Oncon0 4724   dom cdm 4845   ran crn 4846   ` cfv 5423   omcom 6481   ~~ cen 7312   ~<_ cdom 7313   cardccrd 8110   alephcale 8111

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4408  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377  ax-inf2 7852

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-ral 2725  df-rex 2726  df-reu 2727  df-rmo 2728  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-pss 3349  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-tp 3887  df-op 3889  df-uni 4097  df-int 4134  df-iun 4178  df-br 4298  df-opab 4356  df-mpt 4357  df-tr 4391  df-eprel 4637  df-id 4641  df-po 4646  df-so 4647  df-fr 4684  df-se 4685  df-we 4686  df-ord 4727  df-on 4728  df-lim 4729  df-suc 4730  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-isom 5432  df-riota 6057  df-om 6482  df-recs 6837  df-rdg 6871  df-er 7106  df-en 7316  df-dom 7317  df-sdom 7318  df-fin 7319  df-oi 7729  df-har 7778  df-card 8114  df-aleph 8115

This theorem is referenced by: (None)