Theorem pwcdadom 8597

Description: A property of dominance over a powerset, and a main lemma for gchac 9057. 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 8047	. . . 4 |- -. ~P A ~<_* A
2		0elpw 4536	. . . . . . . . . . 11 |- (/) e. ~P ( A +c A )
3		n0i 3709	. . . . . . . . . . 11 |- ( (/) e. ~P ( A +c A ) -> -. ~P ( A +c A ) = (/) )
4	2, 3	ax-mp 5	. . . . . . . . . 10 |- -. ~P ( A +c A ) = (/)
5		dom0 7653	. . . . . . . . . 10 |- ( ~P ( A +c A ) ~<_ (/) <-> ~P ( A +c A ) = (/) )
6	4, 5	mtbir 300	. . . . . . . . 9 |- -. ~P ( A +c A ) ~<_ (/)
7		cdafn 8550	. . . . . . . . . . . 12 |- +c Fn ( _V X. _V )
8		fndm 5636	. . . . . . . . . . . 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 6411	. . . . . . . . . 10 |- ( -. ( A e. _V /\ B e. _V ) -> ( A +c B ) = (/) )
11	10	breq2d 4378	. . . . . . . . 9 |- ( -. ( A e. _V /\ B e. _V ) -> ( ~P ( A +c A ) ~<_ ( A +c B ) <-> ~P ( A +c A ) ~<_ (/) ) )
12	6, 11	mtbiri 304	. . . . . . . 8 |- ( -. ( A e. _V /\ B e. _V ) -> -. ~P ( A +c A ) ~<_ ( A +c B ) )
13	12	con4i 133	. . . . . . 7 |- ( ~P ( A +c A ) ~<_ ( A +c B ) -> ( A e. _V /\ B e. _V ) )
14	13	simpld 460	. . . . . 6 |- ( ~P ( A +c A ) ~<_ ( A +c B ) -> A e. _V )
15		0ex 4499	. . . . . 6 |- (/) e. _V
16		xpsneng 7610	. . . . . 6 |- ( ( A e. _V /\ (/) e. _V ) -> ( A X. { (/) } ) ~~ A )
17	14, 15, 16	sylancl 666	. . . . 5 |- ( ~P ( A +c A ) ~<_ ( A +c B ) -> ( A X. { (/) } ) ~~ A )
18		endom 7550	. . . . 5 |- ( ( A X. { (/) } ) ~~ A -> ( A X. { (/) } ) ~<_ A )
19		domwdom 8042	. . . . 5 |- ( ( A X. { (/) } ) ~<_ A -> ( A X. { (/) } ) ~<_* A )
20		wdomtr 8043	. . . . . 6 |- ( ( ~P A ~<_* ( A X. { (/) } ) /\ ( A X. { (/) } ) ~<_* A ) -> ~P A ~<_* A )
21	20	expcom 436	. . . . 5 |- ( ( A X. { (/) } ) ~<_* A -> ( ~P A ~<_* ( A X. { (/) } ) -> ~P A ~<_* A ) )
22	17, 18, 19, 21	4syl 19	. . . 4 |- ( ~P ( A +c A ) ~<_ ( A +c B ) -> ( ~P A ~<_* ( A X. { (/) } ) -> ~P A ~<_* A ) )
23	1, 22	mtoi 181	. . 3 |- ( ~P ( A +c A ) ~<_ ( A +c B ) -> -. ~P A ~<_* ( A X. { (/) } ) )
24		pwcdaen 8566	. . . . . . . . 9 |- ( ( A e. _V /\ A e. _V ) -> ~P ( A +c A ) ~~ ( ~P A X. ~P A ) )
25	14, 14, 24	syl2anc 665	. . . . . . . 8 |- ( ~P ( A +c A ) ~<_ ( A +c B ) -> ~P ( A +c A ) ~~ ( ~P A X. ~P A ) )
26		domen1 7667	. . . . . . . 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 17	. . . . . . 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 244	. . . . . 6 |- ( ~P ( A +c A ) ~<_ ( A +c B ) -> ( ~P A X. ~P A ) ~<_ ( A +c B ) )
29		cdaval 8551	. . . . . . 7 |- ( ( A e. _V /\ B e. _V ) -> ( A +c B ) = ( ( A X. { (/) } ) u. ( B X. { 1o } ) ) )
30	13, 29	syl 17	. . . . . 6 |- ( ~P ( A +c A ) ~<_ ( A +c B ) -> ( A +c B ) = ( ( A X. { (/) } ) u. ( B X. { 1o } ) ) )
31	28, 30	breqtrd 4391	. . . . 5 |- ( ~P ( A +c A ) ~<_ ( A +c B ) -> ( ~P A X. ~P A ) ~<_ ( ( A X. { (/) } ) u. ( B X. { 1o } ) ) )
32		unxpwdom 8057	. . . . 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 17	. . . 4 |- ( ~P ( A +c A ) ~<_ ( A +c B ) -> ( ~P A ~<_* ( A X. { (/) } ) \/ ~P A ~<_ ( B X. { 1o } ) ) )
34	33	ord 378	. . 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 464	. . 3 |- ( ~P ( A +c A ) ~<_ ( A +c B ) -> B e. _V )
37		1on 7144	. . 3 |- 1o e. On
38		xpsneng 7610	. . 3 |- ( ( B e. _V /\ 1o e. On ) -> ( B X. { 1o } ) ~~ B )
39	36, 37, 38	sylancl 666	. 2 |- ( ~P ( A +c A ) ~<_ ( A +c B ) -> ( B X. { 1o } ) ~~ B )
40		domentr 7582	. 2 |- ( ( ~P A ~<_ ( B X. { 1o } ) /\ ( B X. { 1o } ) ~~ B ) -> ~P A ~<_ B )
41	35, 39, 40	syl2anc 665	1 |- ( ~P ( A +c A ) ~<_ ( A +c B ) -> ~P A ~<_ B )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 187   \/ wo 369   /\ wa 370   = wceq 1437   e. wcel 1872   _Vcvv 3022   u. cun 3377   (/)c0 3704   ~Pcpw 3924   {csn 3941   class class class wbr 4366   X. cxp 4794   dom cdm 4796   Oncon0 5385   Fn wfn 5539  (class class class)co 6249   1oc1o 7130   ~~ cen 7521   ~<_ cdom 7522   ~<_^* cwdom 8025   +c ccda 8548

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2063  ax-ext 2408  ax-sep 4489  ax-nul 4498  ax-pow 4545  ax-pr 4603  ax-un 6541

This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2280  df-mo 2281  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2558  df-ne 2601  df-ral 2719  df-rex 2720  df-rab 2723  df-v 3024  df-sbc 3243  df-csb 3339  df-dif 3382  df-un 3384  df-in 3386  df-ss 3393  df-pss 3395  df-nul 3705  df-if 3855  df-pw 3926  df-sn 3942  df-pr 3944  df-tp 3946  df-op 3948  df-uni 4163  df-int 4199  df-iun 4244  df-br 4367  df-opab 4426  df-mpt 4427  df-tr 4462  df-eprel 4707  df-id 4711  df-po 4717  df-so 4718  df-fr 4755  df-we 4757  df-xp 4802  df-rel 4803  df-cnv 4804  df-co 4805  df-dm 4806  df-rn 4807  df-res 4808  df-ima 4809  df-ord 5388  df-on 5389  df-suc 5391  df-iota 5508  df-fun 5546  df-fn 5547  df-f 5548  df-f1 5549  df-fo 5550  df-f1o 5551  df-fv 5552  df-ov 6252  df-oprab 6253  df-mpt2 6254  df-1st 6751  df-2nd 6752  df-1o 7137  df-2o 7138  df-er 7318  df-map 7429  df-en 7525  df-dom 7526  df-sdom 7527  df-wdom 8027  df-cda 8549

This theorem is referenced by:  gchdomtri  9005  gchpwdom  9046  gchhar  9055