Theorem alephdom2 7973

Description: A dominated initial ordinal is included. (Contributed by Jeff Hankins, 24-Oct-2009.)

Assertion

Ref	Expression
alephdom2	|- ( ( A e. On /\ B e. On ) -> ( ( aleph ` A ) C_ B <-> ( aleph ` A ) ~<_ B ) )

Proof of Theorem alephdom2

Step	Hyp	Ref	Expression
1		alephsdom 7972	. . . 4 |- ( ( B e. On /\ A e. On ) -> ( B e. ( aleph ` A ) <-> B ~< ( aleph ` A ) ) )
2	1	ancoms 441	. . 3 |- ( ( A e. On /\ B e. On ) -> ( B e. ( aleph ` A ) <-> B ~< ( aleph ` A ) ) )
3	2	notbid 287	. 2 |- ( ( A e. On /\ B e. On ) -> ( -. B e. ( aleph ` A ) <-> -. B ~< ( aleph ` A ) ) )
4		alephon 7955	. . . . 5 |- ( aleph ` A ) e. On
5	4	onordi 4689	. . . 4 |- Ord ( aleph ` A )
6		eloni 4594	. . . 4 |- ( B e. On -> Ord B )
7		ordtri1 4617	. . . 4 |- ( ( Ord ( aleph ` A ) /\ Ord B ) -> ( ( aleph ` A ) C_ B <-> -. B e. ( aleph ` A ) ) )
8	5, 6, 7	sylancr 646	. . 3 |- ( B e. On -> ( ( aleph ` A ) C_ B <-> -. B e. ( aleph ` A ) ) )
9	8	adantl 454	. 2 |- ( ( A e. On /\ B e. On ) -> ( ( aleph ` A ) C_ B <-> -. B e. ( aleph ` A ) ) )
10		domtriord 7256	. . . 4 |- ( ( ( aleph ` A ) e. On /\ B e. On ) -> ( ( aleph ` A ) ~<_ B <-> -. B ~< ( aleph ` A ) ) )
11	4, 10	mpan 653	. . 3 |- ( B e. On -> ( ( aleph ` A ) ~<_ B <-> -. B ~< ( aleph ` A ) ) )
12	11	adantl 454	. 2 |- ( ( A e. On /\ B e. On ) -> ( ( aleph ` A ) ~<_ B <-> -. B ~< ( aleph ` A ) ) )
13	3, 9, 12	3bitr4d 278	1 |- ( ( A e. On /\ B e. On ) -> ( ( aleph ` A ) C_ B <-> ( aleph ` A ) ~<_ B ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 178   /\ wa 360   e. wcel 1726   C_ wss 3322   class class class wbr 4215   Ord word 4583   Oncon0 4584   ` cfv 5457   ~<_ cdom 7110   ~< csdm 7111   alephcale 7828

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4323  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704  ax-inf2 7599

This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-tr 4306  df-eprel 4497  df-id 4501  df-po 4506  df-so 4507  df-fr 4544  df-se 4545  df-we 4546  df-ord 4587  df-on 4588  df-lim 4589  df-suc 4590  df-om 4849  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-isom 5466  df-riota 6552  df-recs 6636  df-rdg 6671  df-er 6908  df-en 7113  df-dom 7114  df-sdom 7115  df-fin 7116  df-oi 7482  df-har 7529  df-card 7831  df-aleph 7832