Theorem pwcdadom 8489

Description: A property of dominance over a powerset, and a main lemma for gchac 8952. 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 7898	. . . 4 |- -. ~P A ~<_* A
2		0elpw 4562	. . . . . . . . . . 11 |- (/) e. ~P ( A +c A )
3		n0i 3743	. . . . . . . . . . 11 |- ( (/) e. ~P ( A +c A ) -> -. ~P ( A +c A ) = (/) )
4	2, 3	ax-mp 5	. . . . . . . . . 10 |- -. ~P ( A +c A ) = (/)
5		dom0 7542	. . . . . . . . . 10 |- ( ~P ( A +c A ) ~<_ (/) <-> ~P ( A +c A ) = (/) )
6	4, 5	mtbir 299	. . . . . . . . 9 |- -. ~P ( A +c A ) ~<_ (/)
7		cdafn 8442	. . . . . . . . . . . 12 |- +c Fn ( _V X. _V )
8		fndm 5611	. . . . . . . . . . . 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 6350	. . . . . . . . . 10 |- ( -. ( A e. _V /\ B e. _V ) -> ( A +c B ) = (/) )
11	10	breq2d 4405	. . . . . . . . 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 4523	. . . . . 6 |- (/) e. _V
16		xpsneng 7499	. . . . . 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 7439	. . . . 5 |- ( ( A X. { (/) } ) ~~ A -> ( A X. { (/) } ) ~<_ A )
19		domwdom 7893	. . . . 5 |- ( ( A X. { (/) } ) ~<_ A -> ( A X. { (/) } ) ~<_* A )
20		wdomtr 7894	. . . . . 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 8458	. . . . . . . . 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 7556	. . . . . . . 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 8443	. . . . . . 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 4417	. . . . 5 |- ( ~P ( A +c A ) ~<_ ( A +c B ) -> ( ~P A X. ~P A ) ~<_ ( ( A X. { (/) } ) u. ( B X. { 1o } ) ) )
32		unxpwdom 7908	. . . . 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 7030	. . 3 |- 1o e. On
38		xpsneng 7499	. . 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 7471	. 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 1370   e. wcel 1758   _Vcvv 3071   u. cun 3427   (/)c0 3738   ~Pcpw 3961   {csn 3978   class class class wbr 4393   Oncon0 4820   X. cxp 4939   dom cdm 4941   Fn wfn 5514  (class class class)co 6193   1oc1o 7016   ~~ cen 7410   ~<_ cdom 7411   ~<_^* cwdom 7876   +c ccda 8440

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4514  ax-nul 4522  ax-pow 4571  ax-pr 4632  ax-un 6475

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-rab 2804  df-v 3073  df-sbc 3288  df-csb 3390  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-pss 3445  df-nul 3739  df-if 3893  df-pw 3963  df-sn 3979  df-pr 3981  df-tp 3983  df-op 3985  df-uni 4193  df-int 4230  df-iun 4274  df-br 4394  df-opab 4452  df-mpt 4453  df-tr 4487  df-eprel 4733  df-id 4737  df-po 4742  df-so 4743  df-fr 4780  df-we 4782  df-ord 4823  df-on 4824  df-suc 4826  df-xp 4947  df-rel 4948  df-cnv 4949  df-co 4950  df-dm 4951  df-rn 4952  df-res 4953  df-ima 4954  df-iota 5482  df-fun 5521  df-fn 5522  df-f 5523  df-f1 5524  df-fo 5525  df-f1o 5526  df-fv 5527  df-ov 6196  df-oprab 6197  df-mpt2 6198  df-1st 6680  df-2nd 6681  df-1o 7023  df-2o 7024  df-er 7204  df-map 7319  df-en 7414  df-dom 7415  df-sdom 7416  df-wdom 7878  df-cda 8441

This theorem is referenced by:  gchdomtri  8900  gchpwdom  8941  gchhar  8950