Theorem dfac12k 8969

Description: Equivalence of dfac12 8971 and dfac12a 8970, without using Regularity. (Contributed by Mario Carneiro, 21-May-2015.)

Assertion

Ref	Expression
dfac12k	|- ( A. x e. On ~P x e. dom card <-> A. y e. On ~P ( aleph ` y ) e. dom card )

Distinct variable group:   x, y

Proof of Theorem dfac12k

Step	Hyp	Ref	Expression
1		alephon 8892	. . . 4 |- ( aleph ` y ) e. On
2		pweq 4161	. . . . . 6 |- ( x = ( aleph ` y ) -> ~P x = ~P ( aleph ` y ) )
3	2	eleq1d 2686	. . . . 5 |- ( x = ( aleph ` y ) -> ( ~P x e. dom card <-> ~P ( aleph ` y ) e. dom card ) )
4	3	rspcv 3305	. . . 4 |- ( ( aleph ` y ) e. On -> ( A. x e. On ~P x e. dom card -> ~P ( aleph ` y ) e. dom card ) )
5	1, 4	ax-mp 5	. . 3 |- ( A. x e. On ~P x e. dom card -> ~P ( aleph ` y ) e. dom card )
6	5	ralrimivw 2967	. 2 |- ( A. x e. On ~P x e. dom card -> A. y e. On ~P ( aleph ` y ) e. dom card )
7		omelon 8543	. . . . . . 7 |- om e. On
8		cardon 8770	. . . . . . 7 |- ( card ` x ) e. On
9		ontri1 5757	. . . . . . 7 |- ( ( om e. On /\ ( card ` x ) e. On ) -> ( om C_ ( card ` x ) <-> -. ( card ` x ) e. om ) )
10	7, 8, 9	mp2an 708	. . . . . 6 |- ( om C_ ( card ` x ) <-> -. ( card ` x ) e. om )
11		cardidm 8785	. . . . . . . 8 |- ( card ` ( card ` x ) ) = ( card ` x )
12		cardalephex 8913	. . . . . . . 8 |- ( om C_ ( card ` x ) -> ( ( card ` ( card ` x ) ) = ( card ` x ) <-> E. y e. On ( card ` x ) = ( aleph ` y ) ) )
13	11, 12	mpbii 223	. . . . . . 7 |- ( om C_ ( card ` x ) -> E. y e. On ( card ` x ) = ( aleph ` y ) )
14		r19.29 3072	. . . . . . . . 9 |- ( ( A. y e. On ~P ( aleph ` y ) e. dom card /\ E. y e. On ( card ` x ) = ( aleph ` y ) ) -> E. y e. On ( ~P ( aleph ` y ) e. dom card /\ ( card ` x ) = ( aleph ` y ) ) )
15		pweq 4161	. . . . . . . . . . . 12 |- ( ( card ` x ) = ( aleph ` y ) -> ~P ( card ` x ) = ~P ( aleph ` y ) )
16	15	eleq1d 2686	. . . . . . . . . . 11 |- ( ( card ` x ) = ( aleph ` y ) -> ( ~P ( card ` x ) e. dom card <-> ~P ( aleph ` y ) e. dom card ) )
17	16	biimparc 504	. . . . . . . . . 10 |- ( ( ~P ( aleph ` y ) e. dom card /\ ( card ` x ) = ( aleph ` y ) ) -> ~P ( card ` x ) e. dom card )
18	17	rexlimivw 3029	. . . . . . . . 9 |- ( E. y e. On ( ~P ( aleph ` y ) e. dom card /\ ( card ` x ) = ( aleph ` y ) ) -> ~P ( card ` x ) e. dom card )
19	14, 18	syl 17	. . . . . . . 8 |- ( ( A. y e. On ~P ( aleph ` y ) e. dom card /\ E. y e. On ( card ` x ) = ( aleph ` y ) ) -> ~P ( card ` x ) e. dom card )
20	19	ex 450	. . . . . . 7 |- ( A. y e. On ~P ( aleph ` y ) e. dom card -> ( E. y e. On ( card ` x ) = ( aleph ` y ) -> ~P ( card ` x ) e. dom card ) )
21	13, 20	syl5 34	. . . . . 6 |- ( A. y e. On ~P ( aleph ` y ) e. dom card -> ( om C_ ( card ` x ) -> ~P ( card ` x ) e. dom card ) )
22	10, 21	syl5bir 233	. . . . 5 |- ( A. y e. On ~P ( aleph ` y ) e. dom card -> ( -. ( card ` x ) e. om -> ~P ( card ` x ) e. dom card ) )
23		nnfi 8153	. . . . . . 7 |- ( ( card ` x ) e. om -> ( card ` x ) e. Fin )
24		pwfi 8261	. . . . . . 7 |- ( ( card ` x ) e. Fin <-> ~P ( card ` x ) e. Fin )
25	23, 24	sylib 208	. . . . . 6 |- ( ( card ` x ) e. om -> ~P ( card ` x ) e. Fin )
26		finnum 8774	. . . . . 6 |- ( ~P ( card ` x ) e. Fin -> ~P ( card ` x ) e. dom card )
27	25, 26	syl 17	. . . . 5 |- ( ( card ` x ) e. om -> ~P ( card ` x ) e. dom card )
28	22, 27	pm2.61d2 172	. . . 4 |- ( A. y e. On ~P ( aleph ` y ) e. dom card -> ~P ( card ` x ) e. dom card )
29		oncardid 8782	. . . . 5 |- ( x e. On -> ( card ` x ) ~~ x )
30		pwen 8133	. . . . 5 |- ( ( card ` x ) ~~ x -> ~P ( card ` x ) ~~ ~P x )
31		ennum 8773	. . . . 5 |- ( ~P ( card ` x ) ~~ ~P x -> ( ~P ( card ` x ) e. dom card <-> ~P x e. dom card ) )
32	29, 30, 31	3syl 18	. . . 4 |- ( x e. On -> ( ~P ( card ` x ) e. dom card <-> ~P x e. dom card ) )
33	28, 32	syl5ibcom 235	. . 3 |- ( A. y e. On ~P ( aleph ` y ) e. dom card -> ( x e. On -> ~P x e. dom card ) )
34	33	ralrimiv 2965	. 2 |- ( A. y e. On ~P ( aleph ` y ) e. dom card -> A. x e. On ~P x e. dom card )
35	6, 34	impbii 199	1 |- ( A. x e. On ~P x e. dom card <-> A. y e. On ~P ( aleph ` y ) e. dom card )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   E.wrex 2913   C_ wss 3574   ~Pcpw 4158   class class class wbr 4653   dom cdm 5114   Oncon0 5723   ` cfv 5888   omcom 7065   ~~ cen 7952   Fincfn 7955   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-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-map 7859  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:  dfac12  8971