Theorem pwcdadom 9038

Description: A property of dominance over a powerset, and a main lemma for gchac 9503. 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 8484	. . . 4 |- -. ~P A ~<_* A
2		0elpw 4834	. . . . . . . . . . 11 |- (/) e. ~P ( A +c A )
3	2	n0ii 3922	. . . . . . . . . 10 |- -. ~P ( A +c A ) = (/)
4		dom0 8088	. . . . . . . . . 10 |- ( ~P ( A +c A ) ~<_ (/) <-> ~P ( A +c A ) = (/) )
5	3, 4	mtbir 313	. . . . . . . . 9 |- -. ~P ( A +c A ) ~<_ (/)
6		cdafn 8991	. . . . . . . . . . . 12 |- +c Fn ( _V X. _V )
7		fndm 5990	. . . . . . . . . . . 12 |- ( +c Fn ( _V X. _V ) -> dom +c = ( _V X. _V ) )
8	6, 7	ax-mp 5	. . . . . . . . . . 11 |- dom +c = ( _V X. _V )
9	8	ndmov 6818	. . . . . . . . . 10 |- ( -. ( A e. _V /\ B e. _V ) -> ( A +c B ) = (/) )
10	9	breq2d 4665	. . . . . . . . 9 |- ( -. ( A e. _V /\ B e. _V ) -> ( ~P ( A +c A ) ~<_ ( A +c B ) <-> ~P ( A +c A ) ~<_ (/) ) )
11	5, 10	mtbiri 317	. . . . . . . 8 |- ( -. ( A e. _V /\ B e. _V ) -> -. ~P ( A +c A ) ~<_ ( A +c B ) )
12	11	con4i 113	. . . . . . 7 |- ( ~P ( A +c A ) ~<_ ( A +c B ) -> ( A e. _V /\ B e. _V ) )
13	12	simpld 475	. . . . . 6 |- ( ~P ( A +c A ) ~<_ ( A +c B ) -> A e. _V )
14		0ex 4790	. . . . . 6 |- (/) e. _V
15		xpsneng 8045	. . . . . 6 |- ( ( A e. _V /\ (/) e. _V ) -> ( A X. { (/) } ) ~~ A )
16	13, 14, 15	sylancl 694	. . . . 5 |- ( ~P ( A +c A ) ~<_ ( A +c B ) -> ( A X. { (/) } ) ~~ A )
17		endom 7982	. . . . 5 |- ( ( A X. { (/) } ) ~~ A -> ( A X. { (/) } ) ~<_ A )
18		domwdom 8479	. . . . 5 |- ( ( A X. { (/) } ) ~<_ A -> ( A X. { (/) } ) ~<_* A )
19		wdomtr 8480	. . . . . 6 |- ( ( ~P A ~<_* ( A X. { (/) } ) /\ ( A X. { (/) } ) ~<_* A ) -> ~P A ~<_* A )
20	19	expcom 451	. . . . 5 |- ( ( A X. { (/) } ) ~<_* A -> ( ~P A ~<_* ( A X. { (/) } ) -> ~P A ~<_* A ) )
21	16, 17, 18, 20	4syl 19	. . . 4 |- ( ~P ( A +c A ) ~<_ ( A +c B ) -> ( ~P A ~<_* ( A X. { (/) } ) -> ~P A ~<_* A ) )
22	1, 21	mtoi 190	. . 3 |- ( ~P ( A +c A ) ~<_ ( A +c B ) -> -. ~P A ~<_* ( A X. { (/) } ) )
23		pwcdaen 9007	. . . . . . . . 9 |- ( ( A e. _V /\ A e. _V ) -> ~P ( A +c A ) ~~ ( ~P A X. ~P A ) )
24	13, 13, 23	syl2anc 693	. . . . . . . 8 |- ( ~P ( A +c A ) ~<_ ( A +c B ) -> ~P ( A +c A ) ~~ ( ~P A X. ~P A ) )
25		domen1 8102	. . . . . . . 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 ) ) )
26	24, 25	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 ) ) )
27	26	ibi 256	. . . . . 6 |- ( ~P ( A +c A ) ~<_ ( A +c B ) -> ( ~P A X. ~P A ) ~<_ ( A +c B ) )
28		cdaval 8992	. . . . . . 7 |- ( ( A e. _V /\ B e. _V ) -> ( A +c B ) = ( ( A X. { (/) } ) u. ( B X. { 1o } ) ) )
29	12, 28	syl 17	. . . . . 6 |- ( ~P ( A +c A ) ~<_ ( A +c B ) -> ( A +c B ) = ( ( A X. { (/) } ) u. ( B X. { 1o } ) ) )
30	27, 29	breqtrd 4679	. . . . 5 |- ( ~P ( A +c A ) ~<_ ( A +c B ) -> ( ~P A X. ~P A ) ~<_ ( ( A X. { (/) } ) u. ( B X. { 1o } ) ) )
31		unxpwdom 8494	. . . . 5 |- ( ( ~P A X. ~P A ) ~<_ ( ( A X. { (/) } ) u. ( B X. { 1o } ) ) -> ( ~P A ~<_* ( A X. { (/) } ) \/ ~P A ~<_ ( B X. { 1o } ) ) )
32	30, 31	syl 17	. . . 4 |- ( ~P ( A +c A ) ~<_ ( A +c B ) -> ( ~P A ~<_* ( A X. { (/) } ) \/ ~P A ~<_ ( B X. { 1o } ) ) )
33	32	ord 392	. . 3 |- ( ~P ( A +c A ) ~<_ ( A +c B ) -> ( -. ~P A ~<_* ( A X. { (/) } ) -> ~P A ~<_ ( B X. { 1o } ) ) )
34	22, 33	mpd 15	. 2 |- ( ~P ( A +c A ) ~<_ ( A +c B ) -> ~P A ~<_ ( B X. { 1o } ) )
35	12	simprd 479	. . 3 |- ( ~P ( A +c A ) ~<_ ( A +c B ) -> B e. _V )
36		1on 7567	. . 3 |- 1o e. On
37		xpsneng 8045	. . 3 |- ( ( B e. _V /\ 1o e. On ) -> ( B X. { 1o } ) ~~ B )
38	35, 36, 37	sylancl 694	. 2 |- ( ~P ( A +c A ) ~<_ ( A +c B ) -> ( B X. { 1o } ) ~~ B )
39		domentr 8015	. 2 |- ( ( ~P A ~<_ ( B X. { 1o } ) /\ ( B X. { 1o } ) ~~ B ) -> ~P A ~<_ B )
40	34, 38, 39	syl2anc 693	1 |- ( ~P ( A +c A ) ~<_ ( A +c B ) -> ~P A ~<_ B )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   u. cun 3572   (/)c0 3915   ~Pcpw 4158   {csn 4177   class class class wbr 4653   X. cxp 5112   dom cdm 5114   Oncon0 5723   Fn wfn 5883  (class class class)co 6650   1oc1o 7553   ~~ cen 7952   ~<_ cdom 7953   ~<_^* cwdom 8462   +c ccda 8989

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-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

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-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-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-ord 5726  df-on 5727  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-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-1o 7560  df-2o 7561  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-wdom 8464  df-cda 8990

This theorem is referenced by:  gchdomtri  9451  gchpwdom  9492  gchhar  9501