Theorem alephinit 8523

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 8520	. . . . 5 |- ( ( om C_ A /\ ( card ` A ) = A ) <-> A e. ran aleph )
2	1	bicomi 206	. . . 4 |- ( A e. ran aleph <-> ( om C_ A /\ ( card ` A ) = A ) )
3	2	baib 913	. . 3 |- ( om C_ A -> ( A e. ran aleph <-> ( card ` A ) = A ) )
4	3	adantl 468	. 2 |- ( ( A e. On /\ om C_ A ) -> ( A e. ran aleph <-> ( card ` A ) = A ) )
5		onenon 8380	. . . . . . . 8 |- ( A e. On -> A e. dom card )
6	5	adantr 467	. . . . . . 7 |- ( ( A e. On /\ om C_ A ) -> A e. dom card )
7		onenon 8380	. . . . . . 7 |- ( x e. On -> x e. dom card )
8		carddom2 8408	. . . . . . 7 |- ( ( A e. dom card /\ x e. dom card ) -> ( ( card ` A ) C_ ( card ` x ) <-> A ~<_ x ) )
9	6, 7, 8	syl2an 480	. . . . . 6 |- ( ( ( A e. On /\ om C_ A ) /\ x e. On ) -> ( ( card ` A ) C_ ( card ` x ) <-> A ~<_ x ) )
10		cardonle 8388	. . . . . . . 8 |- ( x e. On -> ( card ` x ) C_ x )
11	10	adantl 468	. . . . . . 7 |- ( ( ( A e. On /\ om C_ A ) /\ x e. On ) -> ( card ` x ) C_ x )
12		sstr 3439	. . . . . . . 8 |- ( ( ( card ` A ) C_ ( card ` x ) /\ ( card ` x ) C_ x ) -> ( card ` A ) C_ x )
13	12	expcom 437	. . . . . . 7 |- ( ( card ` x ) C_ x -> ( ( card ` A ) C_ ( card ` x ) -> ( card ` A ) C_ x ) )
14	11, 13	syl 17	. . . . . 6 |- ( ( ( A e. On /\ om C_ A ) /\ x e. On ) -> ( ( card ` A ) C_ ( card ` x ) -> ( card ` A ) C_ x ) )
15	9, 14	sylbird 239	. . . . 5 |- ( ( ( A e. On /\ om C_ A ) /\ x e. On ) -> ( A ~<_ x -> ( card ` A ) C_ x ) )
16		sseq1 3452	. . . . . 6 |- ( ( card ` A ) = A -> ( ( card ` A ) C_ x <-> A C_ x ) )
17	16	imbi2d 318	. . . . 5 |- ( ( card ` A ) = A -> ( ( A ~<_ x -> ( card ` A ) C_ x ) <-> ( A ~<_ x -> A C_ x ) ) )
18	15, 17	syl5ibcom 224	. . . 4 |- ( ( ( A e. On /\ om C_ A ) /\ x e. On ) -> ( ( card ` A ) = A -> ( A ~<_ x -> A C_ x ) ) )
19	18	ralrimdva 2805	. . 3 |- ( ( A e. On /\ om C_ A ) -> ( ( card ` A ) = A -> A. x e. On ( A ~<_ x -> A C_ x ) ) )
20		oncardid 8387	. . . . . . 7 |- ( A e. On -> ( card ` A ) ~~ A )
21		ensym 7615	. . . . . . 7 |- ( ( card ` A ) ~~ A -> A ~~ ( card ` A ) )
22		endom 7593	. . . . . . 7 |- ( A ~~ ( card ` A ) -> A ~<_ ( card ` A ) )
23	20, 21, 22	3syl 18	. . . . . 6 |- ( A e. On -> A ~<_ ( card ` A ) )
24	23	adantr 467	. . . . 5 |- ( ( A e. On /\ om C_ A ) -> A ~<_ ( card ` A ) )
25		cardon 8375	. . . . . 6 |- ( card ` A ) e. On
26		breq2 4405	. . . . . . . 8 |- ( x = ( card ` A ) -> ( A ~<_ x <-> A ~<_ ( card ` A ) ) )
27		sseq2 3453	. . . . . . . 8 |- ( x = ( card ` A ) -> ( A C_ x <-> A C_ ( card ` A ) ) )
28	26, 27	imbi12d 322	. . . . . . 7 |- ( x = ( card ` A ) -> ( ( A ~<_ x -> A C_ x ) <-> ( A ~<_ ( card ` A ) -> A C_ ( card ` A ) ) ) )
29	28	rspcv 3145	. . . . . 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 31	. . . 4 |- ( ( A e. On /\ om C_ A ) -> ( A. x e. On ( A ~<_ x -> A C_ x ) -> A C_ ( card ` A ) ) )
32		cardonle 8388	. . . . . . 7 |- ( A e. On -> ( card ` A ) C_ A )
33	32	adantr 467	. . . . . 6 |- ( ( A e. On /\ om C_ A ) -> ( card ` A ) C_ A )
34	33	biantrurd 511	. . . . 5 |- ( ( A e. On /\ om C_ A ) -> ( A C_ ( card ` A ) <-> ( ( card ` A ) C_ A /\ A C_ ( card ` A ) ) ) )
35		eqss 3446	. . . . 5 |- ( ( card ` A ) = A <-> ( ( card ` A ) C_ A /\ A C_ ( card ` A ) ) )
36	34, 35	syl6bbr 267	. . . 4 |- ( ( A e. On /\ om C_ A ) -> ( A C_ ( card ` A ) <-> ( card ` A ) = A ) )
37	31, 36	sylibd 218	. . 3 |- ( ( A e. On /\ om C_ A ) -> ( A. x e. On ( A ~<_ x -> A C_ x ) -> ( card ` A ) = A ) )
38	19, 37	impbid 194	. 2 |- ( ( A e. On /\ om C_ A ) -> ( ( card ` A ) = A <-> A. x e. On ( A ~<_ x -> A C_ x ) ) )
39	4, 38	bitrd 257	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 188   /\ wa 371   = wceq 1443   e. wcel 1886   A.wral 2736   C_ wss 3403   class class class wbr 4401   dom cdm 4833   ran crn 4834   Oncon0 5422   ` cfv 5581   omcom 6689   ~~ cen 7563   ~<_ cdom 7564   cardccrd 8366   alephcale 8367

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1668  ax-4 1681  ax-5 1757  ax-6 1804  ax-7 1850  ax-8 1888  ax-9 1895  ax-10 1914  ax-11 1919  ax-12 1932  ax-13 2090  ax-ext 2430  ax-rep 4514  ax-sep 4524  ax-nul 4533  ax-pow 4580  ax-pr 4638  ax-un 6580  ax-inf2 8143

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 985  df-3an 986  df-tru 1446  df-ex 1663  df-nf 1667  df-sb 1797  df-eu 2302  df-mo 2303  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2580  df-ne 2623  df-ral 2741  df-rex 2742  df-reu 2743  df-rmo 2744  df-rab 2745  df-v 3046  df-sbc 3267  df-csb 3363  df-dif 3406  df-un 3408  df-in 3410  df-ss 3417  df-pss 3419  df-nul 3731  df-if 3881  df-pw 3952  df-sn 3968  df-pr 3970  df-tp 3972  df-op 3974  df-uni 4198  df-int 4234  df-iun 4279  df-br 4402  df-opab 4461  df-mpt 4462  df-tr 4497  df-eprel 4744  df-id 4748  df-po 4754  df-so 4755  df-fr 4792  df-se 4793  df-we 4794  df-xp 4839  df-rel 4840  df-cnv 4841  df-co 4842  df-dm 4843  df-rn 4844  df-res 4845  df-ima 4846  df-pred 5379  df-ord 5425  df-on 5426  df-lim 5427  df-suc 5428  df-iota 5545  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-isom 5590  df-riota 6250  df-om 6690  df-wrecs 7025  df-recs 7087  df-rdg 7125  df-er 7360  df-en 7567  df-dom 7568  df-sdom 7569  df-fin 7570  df-oi 8022  df-har 8070  df-card 8370  df-aleph 8371

This theorem is referenced by: (None)