Theorem alephinit 8918

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 8915	. . . . 5 |- ( ( om C_ A /\ ( card ` A ) = A ) <-> A e. ran aleph )
2	1	bicomi 214	. . . 4 |- ( A e. ran aleph <-> ( om C_ A /\ ( card ` A ) = A ) )
3	2	baib 944	. . 3 |- ( om C_ A -> ( A e. ran aleph <-> ( card ` A ) = A ) )
4	3	adantl 482	. 2 |- ( ( A e. On /\ om C_ A ) -> ( A e. ran aleph <-> ( card ` A ) = A ) )
5		onenon 8775	. . . . . . . 8 |- ( A e. On -> A e. dom card )
6	5	adantr 481	. . . . . . 7 |- ( ( A e. On /\ om C_ A ) -> A e. dom card )
7		onenon 8775	. . . . . . 7 |- ( x e. On -> x e. dom card )
8		carddom2 8803	. . . . . . 7 |- ( ( A e. dom card /\ x e. dom card ) -> ( ( card ` A ) C_ ( card ` x ) <-> A ~<_ x ) )
9	6, 7, 8	syl2an 494	. . . . . 6 |- ( ( ( A e. On /\ om C_ A ) /\ x e. On ) -> ( ( card ` A ) C_ ( card ` x ) <-> A ~<_ x ) )
10		cardonle 8783	. . . . . . . 8 |- ( x e. On -> ( card ` x ) C_ x )
11	10	adantl 482	. . . . . . 7 |- ( ( ( A e. On /\ om C_ A ) /\ x e. On ) -> ( card ` x ) C_ x )
12		sstr 3611	. . . . . . . 8 |- ( ( ( card ` A ) C_ ( card ` x ) /\ ( card ` x ) C_ x ) -> ( card ` A ) C_ x )
13	12	expcom 451	. . . . . . 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 250	. . . . 5 |- ( ( ( A e. On /\ om C_ A ) /\ x e. On ) -> ( A ~<_ x -> ( card ` A ) C_ x ) )
16		sseq1 3626	. . . . . 6 |- ( ( card ` A ) = A -> ( ( card ` A ) C_ x <-> A C_ x ) )
17	16	imbi2d 330	. . . . 5 |- ( ( card ` A ) = A -> ( ( A ~<_ x -> ( card ` A ) C_ x ) <-> ( A ~<_ x -> A C_ x ) ) )
18	15, 17	syl5ibcom 235	. . . 4 |- ( ( ( A e. On /\ om C_ A ) /\ x e. On ) -> ( ( card ` A ) = A -> ( A ~<_ x -> A C_ x ) ) )
19	18	ralrimdva 2969	. . 3 |- ( ( A e. On /\ om C_ A ) -> ( ( card ` A ) = A -> A. x e. On ( A ~<_ x -> A C_ x ) ) )
20		oncardid 8782	. . . . . . 7 |- ( A e. On -> ( card ` A ) ~~ A )
21		ensym 8005	. . . . . . 7 |- ( ( card ` A ) ~~ A -> A ~~ ( card ` A ) )
22		endom 7982	. . . . . . 7 |- ( A ~~ ( card ` A ) -> A ~<_ ( card ` A ) )
23	20, 21, 22	3syl 18	. . . . . 6 |- ( A e. On -> A ~<_ ( card ` A ) )
24	23	adantr 481	. . . . 5 |- ( ( A e. On /\ om C_ A ) -> A ~<_ ( card ` A ) )
25		cardon 8770	. . . . . 6 |- ( card ` A ) e. On
26		breq2 4657	. . . . . . . 8 |- ( x = ( card ` A ) -> ( A ~<_ x <-> A ~<_ ( card ` A ) ) )
27		sseq2 3627	. . . . . . . 8 |- ( x = ( card ` A ) -> ( A C_ x <-> A C_ ( card ` A ) ) )
28	26, 27	imbi12d 334	. . . . . . 7 |- ( x = ( card ` A ) -> ( ( A ~<_ x -> A C_ x ) <-> ( A ~<_ ( card ` A ) -> A C_ ( card ` A ) ) ) )
29	28	rspcv 3305	. . . . . 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 8783	. . . . . . 7 |- ( A e. On -> ( card ` A ) C_ A )
33	32	adantr 481	. . . . . 6 |- ( ( A e. On /\ om C_ A ) -> ( card ` A ) C_ A )
34	33	biantrurd 529	. . . . 5 |- ( ( A e. On /\ om C_ A ) -> ( A C_ ( card ` A ) <-> ( ( card ` A ) C_ A /\ A C_ ( card ` A ) ) ) )
35		eqss 3618	. . . . 5 |- ( ( card ` A ) = A <-> ( ( card ` A ) C_ A /\ A C_ ( card ` A ) ) )
36	34, 35	syl6bbr 278	. . . 4 |- ( ( A e. On /\ om C_ A ) -> ( A C_ ( card ` A ) <-> ( card ` A ) = A ) )
37	31, 36	sylibd 229	. . 3 |- ( ( A e. On /\ om C_ A ) -> ( A. x e. On ( A ~<_ x -> A C_ x ) -> ( card ` A ) = A ) )
38	19, 37	impbid 202	. 2 |- ( ( A e. On /\ om C_ A ) -> ( ( card ` A ) = A <-> A. x e. On ( A ~<_ x -> A C_ x ) ) )
39	4, 38	bitrd 268	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 196   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   C_ wss 3574   class class class wbr 4653   dom cdm 5114   ran crn 5115   Oncon0 5723   ` cfv 5888   omcom 7065   ~~ cen 7952   ~<_ cdom 7953   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: (None)