Theorem pwdom 8112

Description: Injection of sets implies injection on power sets. (Contributed by Mario Carneiro, 9-Apr-2015.)

Assertion

Ref	Expression
pwdom	|- ( A ~<_ B -> ~P A ~<_ ~P B )

Proof of Theorem pwdom

Dummy variable f is distinct from all other variables.

Step	Hyp	Ref	Expression
1		pweq 4161	. . 3 |- ( A = (/) -> ~P A = ~P (/) )
2	1	breq1d 4663	. 2 |- ( A = (/) -> ( ~P A ~<_ ~P B <-> ~P (/) ~<_ ~P B ) )
3		reldom 7961	. . . . . . 7 |- Rel ~<_
4	3	brrelexi 5158	. . . . . 6 |- ( A ~<_ B -> A e. _V )
5		0sdomg 8089	. . . . . 6 |- ( A e. _V -> ( (/) ~< A <-> A =/= (/) ) )
6	4, 5	syl 17	. . . . 5 |- ( A ~<_ B -> ( (/) ~< A <-> A =/= (/) ) )
7	6	biimpar 502	. . . 4 |- ( ( A ~<_ B /\ A =/= (/) ) -> (/) ~< A )
8		simpl 473	. . . 4 |- ( ( A ~<_ B /\ A =/= (/) ) -> A ~<_ B )
9		fodomr 8111	. . . 4 |- ( ( (/) ~< A /\ A ~<_ B ) -> E. f f : B -onto-> A )
10	7, 8, 9	syl2anc 693	. . 3 |- ( ( A ~<_ B /\ A =/= (/) ) -> E. f f : B -onto-> A )
11		vex 3203	. . . . 5 |- f e. _V
12		fopwdom 8068	. . . . 5 |- ( ( f e. _V /\ f : B -onto-> A ) -> ~P A ~<_ ~P B )
13	11, 12	mpan 706	. . . 4 |- ( f : B -onto-> A -> ~P A ~<_ ~P B )
14	13	exlimiv 1858	. . 3 |- ( E. f f : B -onto-> A -> ~P A ~<_ ~P B )
15	10, 14	syl 17	. 2 |- ( ( A ~<_ B /\ A =/= (/) ) -> ~P A ~<_ ~P B )
16	3	brrelex2i 5159	. . . 4 |- ( A ~<_ B -> B e. _V )
17		pwexg 4850	. . . 4 |- ( B e. _V -> ~P B e. _V )
18	16, 17	syl 17	. . 3 |- ( A ~<_ B -> ~P B e. _V )
19		0ss 3972	. . . 4 |- (/) C_ B
20		sspwb 4917	. . . 4 |- ( (/) C_ B <-> ~P (/) C_ ~P B )
21	19, 20	mpbi 220	. . 3 |- ~P (/) C_ ~P B
22		ssdomg 8001	. . 3 |- ( ~P B e. _V -> ( ~P (/) C_ ~P B -> ~P (/) ~<_ ~P B ) )
23	18, 21, 22	mpisyl 21	. 2 |- ( A ~<_ B -> ~P (/) ~<_ ~P B )
24	2, 15, 23	pm2.61ne 2879	1 |- ( A ~<_ B -> ~P A ~<_ ~P B )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   E.wex 1704   e. wcel 1990   =/= wne 2794   _Vcvv 3200   C_ wss 3574   (/)c0 3915   ~Pcpw 4158   class class class wbr 4653   -onto->wfo 5886   ~<_ cdom 7953   ~< csdm 7954

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-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-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  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-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958

This theorem is referenced by:  cdalepw  9018  gchpwdom  9492  gchaclem  9500  2ndcredom  21253