Theorem alephdom2 7928

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 7927	. . . 4 |- ( ( B e. On /\ A e. On ) -> ( B e. ( aleph ` A ) <-> B ~< ( aleph ` A ) ) )
2	1	ancoms 440	. . 3 |- ( ( A e. On /\ B e. On ) -> ( B e. ( aleph ` A ) <-> B ~< ( aleph ` A ) ) )
3	2	notbid 286	. 2 |- ( ( A e. On /\ B e. On ) -> ( -. B e. ( aleph ` A ) <-> -. B ~< ( aleph ` A ) ) )
4		alephon 7910	. . . . 5 |- ( aleph ` A ) e. On
5	4	onordi 4649	. . . 4 |- Ord ( aleph ` A )
6		eloni 4555	. . . 4 |- ( B e. On -> Ord B )
7		ordtri1 4578	. . . 4 |- ( ( Ord ( aleph ` A ) /\ Ord B ) -> ( ( aleph ` A ) C_ B <-> -. B e. ( aleph ` A ) ) )
8	5, 6, 7	sylancr 645	. . 3 |- ( B e. On -> ( ( aleph ` A ) C_ B <-> -. B e. ( aleph ` A ) ) )
9	8	adantl 453	. 2 |- ( ( A e. On /\ B e. On ) -> ( ( aleph ` A ) C_ B <-> -. B e. ( aleph ` A ) ) )
10		domtriord 7216	. . . 4 |- ( ( ( aleph ` A ) e. On /\ B e. On ) -> ( ( aleph ` A ) ~<_ B <-> -. B ~< ( aleph ` A ) ) )
11	4, 10	mpan 652	. . 3 |- ( B e. On -> ( ( aleph ` A ) ~<_ B <-> -. B ~< ( aleph ` A ) ) )
12	11	adantl 453	. 2 |- ( ( A e. On /\ B e. On ) -> ( ( aleph ` A ) ~<_ B <-> -. B ~< ( aleph ` A ) ) )
13	3, 9, 12	3bitr4d 277	1 |- ( ( A e. On /\ B e. On ) -> ( ( aleph ` A ) C_ B <-> ( aleph ` A ) ~<_ B ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 177   /\ wa 359   e. wcel 1721   C_ wss 3284   class class class wbr 4176   Ord word 4544   Oncon0 4545   ` cfv 5417   ~<_ cdom 7070   ~< csdm 7071   alephcale 7783

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389  ax-rep 4284  ax-sep 4294  ax-nul 4302  ax-pow 4341  ax-pr 4367  ax-un 4664  ax-inf2 7556

This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-ral 2675  df-rex 2676  df-reu 2677  df-rmo 2678  df-rab 2679  df-v 2922  df-sbc 3126  df-csb 3216  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-pss 3300  df-nul 3593  df-if 3704  df-pw 3765  df-sn 3784  df-pr 3785  df-tp 3786  df-op 3787  df-uni 3980  df-int 4015  df-iun 4059  df-br 4177  df-opab 4231  df-mpt 4232  df-tr 4267  df-eprel 4458  df-id 4462  df-po 4467  df-so 4468  df-fr 4505  df-se 4506  df-we 4507  df-ord 4548  df-on 4549  df-lim 4550  df-suc 4551  df-om 4809  df-xp 4847  df-rel 4848  df-cnv 4849  df-co 4850  df-dm 4851  df-rn 4852  df-res 4853  df-ima 4854  df-iota 5381  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-isom 5426  df-riota 6512  df-recs 6596  df-rdg 6631  df-er 6868  df-en 7073  df-dom 7074  df-sdom 7075  df-fin 7076  df-oi 7439  df-har 7486  df-card 7786  df-aleph 7787