Theorem pwcdadom 8608

Description: A property of dominance over a powerset, and a main lemma for gchac 9071. Similar to Lemma 2.3 of [KanamoriPincus] p. 420. (Contributed by Mario Carneiro, 15-May-2015.)

Assertion

Ref	Expression
pwcdadom	|- ( ~P ( A +c A ) ~<_ ( A +c B ) -> ~P A ~<_ B )

Proof of Theorem pwcdadom

Step	Hyp	Ref	Expression
1		canthwdom 8017	. . . 4 |- -. ~P A ~<_* A
2		0elpw 4622	. . . . . . . . . . 11 |- (/) e. ~P ( A +c A )
3		n0i 3795	. . . . . . . . . . 11 |- ( (/) e. ~P ( A +c A ) -> -. ~P ( A +c A ) = (/) )
4	2, 3	ax-mp 5	. . . . . . . . . 10 |- -. ~P ( A +c A ) = (/)
5		dom0 7657	. . . . . . . . . 10 |- ( ~P ( A +c A ) ~<_ (/) <-> ~P ( A +c A ) = (/) )
6	4, 5	mtbir 299	. . . . . . . . 9 |- -. ~P ( A +c A ) ~<_ (/)
7		cdafn 8561	. . . . . . . . . . . 12 |- +c Fn ( _V X. _V )
8		fndm 5686	. . . . . . . . . . . 12 |- ( +c Fn ( _V X. _V ) -> dom +c = ( _V X. _V ) )
9	7, 8	ax-mp 5	. . . . . . . . . . 11 |- dom +c = ( _V X. _V )
10	9	ndmov 6454	. . . . . . . . . 10 |- ( -. ( A e. _V /\ B e. _V ) -> ( A +c B ) = (/) )
11	10	breq2d 4465	. . . . . . . . 9 |- ( -. ( A e. _V /\ B e. _V ) -> ( ~P ( A +c A ) ~<_ ( A +c B ) <-> ~P ( A +c A ) ~<_ (/) ) )
12	6, 11	mtbiri 303	. . . . . . . 8 |- ( -. ( A e. _V /\ B e. _V ) -> -. ~P ( A +c A ) ~<_ ( A +c B ) )
13	12	con4i 130	. . . . . . 7 |- ( ~P ( A +c A ) ~<_ ( A +c B ) -> ( A e. _V /\ B e. _V ) )
14	13	simpld 459	. . . . . 6 |- ( ~P ( A +c A ) ~<_ ( A +c B ) -> A e. _V )
15		0ex 4583	. . . . . 6 |- (/) e. _V
16		xpsneng 7614	. . . . . 6 |- ( ( A e. _V /\ (/) e. _V ) -> ( A X. { (/) } ) ~~ A )
17	14, 15, 16	sylancl 662	. . . . 5 |- ( ~P ( A +c A ) ~<_ ( A +c B ) -> ( A X. { (/) } ) ~~ A )
18		endom 7554	. . . . 5 |- ( ( A X. { (/) } ) ~~ A -> ( A X. { (/) } ) ~<_ A )
19		domwdom 8012	. . . . 5 |- ( ( A X. { (/) } ) ~<_ A -> ( A X. { (/) } ) ~<_* A )
20		wdomtr 8013	. . . . . 6 |- ( ( ~P A ~<_* ( A X. { (/) } ) /\ ( A X. { (/) } ) ~<_* A ) -> ~P A ~<_* A )
21	20	expcom 435	. . . . 5 |- ( ( A X. { (/) } ) ~<_* A -> ( ~P A ~<_* ( A X. { (/) } ) -> ~P A ~<_* A ) )
22	17, 18, 19, 21	4syl 21	. . . 4 |- ( ~P ( A +c A ) ~<_ ( A +c B ) -> ( ~P A ~<_* ( A X. { (/) } ) -> ~P A ~<_* A ) )
23	1, 22	mtoi 178	. . 3 |- ( ~P ( A +c A ) ~<_ ( A +c B ) -> -. ~P A ~<_* ( A X. { (/) } ) )
24		pwcdaen 8577	. . . . . . . . 9 |- ( ( A e. _V /\ A e. _V ) -> ~P ( A +c A ) ~~ ( ~P A X. ~P A ) )
25	14, 14, 24	syl2anc 661	. . . . . . . 8 |- ( ~P ( A +c A ) ~<_ ( A +c B ) -> ~P ( A +c A ) ~~ ( ~P A X. ~P A ) )
26		domen1 7671	. . . . . . . 8 |- ( ~P ( A +c A ) ~~ ( ~P A X. ~P A ) -> ( ~P ( A +c A ) ~<_ ( A +c B ) <-> ( ~P A X. ~P A ) ~<_ ( A +c B ) ) )
27	25, 26	syl 16	. . . . . . 7 |- ( ~P ( A +c A ) ~<_ ( A +c B ) -> ( ~P ( A +c A ) ~<_ ( A +c B ) <-> ( ~P A X. ~P A ) ~<_ ( A +c B ) ) )
28	27	ibi 241	. . . . . 6 |- ( ~P ( A +c A ) ~<_ ( A +c B ) -> ( ~P A X. ~P A ) ~<_ ( A +c B ) )
29		cdaval 8562	. . . . . . 7 |- ( ( A e. _V /\ B e. _V ) -> ( A +c B ) = ( ( A X. { (/) } ) u. ( B X. { 1o } ) ) )
30	13, 29	syl 16	. . . . . 6 |- ( ~P ( A +c A ) ~<_ ( A +c B ) -> ( A +c B ) = ( ( A X. { (/) } ) u. ( B X. { 1o } ) ) )
31	28, 30	breqtrd 4477	. . . . 5 |- ( ~P ( A +c A ) ~<_ ( A +c B ) -> ( ~P A X. ~P A ) ~<_ ( ( A X. { (/) } ) u. ( B X. { 1o } ) ) )
32		unxpwdom 8027	. . . . 5 |- ( ( ~P A X. ~P A ) ~<_ ( ( A X. { (/) } ) u. ( B X. { 1o } ) ) -> ( ~P A ~<_* ( A X. { (/) } ) \/ ~P A ~<_ ( B X. { 1o } ) ) )
33	31, 32	syl 16	. . . 4 |- ( ~P ( A +c A ) ~<_ ( A +c B ) -> ( ~P A ~<_* ( A X. { (/) } ) \/ ~P A ~<_ ( B X. { 1o } ) ) )
34	33	ord 377	. . 3 |- ( ~P ( A +c A ) ~<_ ( A +c B ) -> ( -. ~P A ~<_* ( A X. { (/) } ) -> ~P A ~<_ ( B X. { 1o } ) ) )
35	23, 34	mpd 15	. 2 |- ( ~P ( A +c A ) ~<_ ( A +c B ) -> ~P A ~<_ ( B X. { 1o } ) )
36	13	simprd 463	. . 3 |- ( ~P ( A +c A ) ~<_ ( A +c B ) -> B e. _V )
37		1on 7149	. . 3 |- 1o e. On
38		xpsneng 7614	. . 3 |- ( ( B e. _V /\ 1o e. On ) -> ( B X. { 1o } ) ~~ B )
39	36, 37, 38	sylancl 662	. 2 |- ( ~P ( A +c A ) ~<_ ( A +c B ) -> ( B X. { 1o } ) ~~ B )
40		domentr 7586	. 2 |- ( ( ~P A ~<_ ( B X. { 1o } ) /\ ( B X. { 1o } ) ~~ B ) -> ~P A ~<_ B )
41	35, 39, 40	syl2anc 661	1 |- ( ~P ( A +c A ) ~<_ ( A +c B ) -> ~P A ~<_ B )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 184   \/ wo 368   /\ wa 369   = wceq 1379   e. wcel 1767   _Vcvv 3118   u. cun 3479   (/)c0 3790   ~Pcpw 4016   {csn 4033   class class class wbr 4453   Oncon0 4884   X. cxp 5003   dom cdm 5005   Fn wfn 5589  (class class class)co 6295   1oc1o 7135   ~~ cen 7525   ~<_ cdom 7526   ~<_^* cwdom 7995   +c ccda 8559

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-int 4289  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-we 4846  df-ord 4887  df-on 4888  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-1st 6795  df-2nd 6796  df-1o 7142  df-2o 7143  df-er 7323  df-map 7434  df-en 7529  df-dom 7530  df-sdom 7531  df-wdom 7997  df-cda 8560

This theorem is referenced by:  gchdomtri  9019  gchpwdom  9060  gchhar  9069