Theorem pwcda1 9016

Description: The sum of a powerset with itself is equipotent to the successor powerset. (Contributed by Mario Carneiro, 15-May-2015.)

Assertion

Ref	Expression
pwcda1	|- ( A e. V -> ( ~P A +c ~P A ) ~~ ~P ( A +c 1o ) )

Proof of Theorem pwcda1

Step	Hyp	Ref	Expression
1		1on 7567	. . . 4 |- 1o e. On
2		pwcdaen 9007	. . . 4 |- ( ( A e. V /\ 1o e. On ) -> ~P ( A +c 1o ) ~~ ( ~P A X. ~P 1o ) )
3	1, 2	mpan2 707	. . 3 |- ( A e. V -> ~P ( A +c 1o ) ~~ ( ~P A X. ~P 1o ) )
4		pwpw0 4344	. . . . . 6 |- ~P { (/) } = { (/) , { (/) } }
5		df1o2 7572	. . . . . . 7 |- 1o = { (/) }
6	5	pweqi 4162	. . . . . 6 |- ~P 1o = ~P { (/) }
7		df2o2 7574	. . . . . 6 |- 2o = { (/) , { (/) } }
8	4, 6, 7	3eqtr4i 2654	. . . . 5 |- ~P 1o = 2o
9	8	xpeq2i 5136	. . . 4 |- ( ~P A X. ~P 1o ) = ( ~P A X. 2o )
10		pwexg 4850	. . . . 5 |- ( A e. V -> ~P A e. _V )
11		xp2cda 9002	. . . . 5 |- ( ~P A e. _V -> ( ~P A X. 2o ) = ( ~P A +c ~P A ) )
12	10, 11	syl 17	. . . 4 |- ( A e. V -> ( ~P A X. 2o ) = ( ~P A +c ~P A ) )
13	9, 12	syl5eq 2668	. . 3 |- ( A e. V -> ( ~P A X. ~P 1o ) = ( ~P A +c ~P A ) )
14	3, 13	breqtrd 4679	. 2 |- ( A e. V -> ~P ( A +c 1o ) ~~ ( ~P A +c ~P A ) )
15	14	ensymd 8007	1 |- ( A e. V -> ( ~P A +c ~P A ) ~~ ~P ( A +c 1o ) )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   _Vcvv 3200   (/)c0 3915   ~Pcpw 4158   {csn 4177   {cpr 4179   class class class wbr 4653   X. cxp 5112   Oncon0 5723  (class class class)co 6650   1oc1o 7553   2oc2o 7554   ~~ cen 7952   +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-cda 8990

This theorem is referenced by:  pwcdaidm  9017  cdalepw  9018  pwsdompw  9026  gchcdaidm  9490  gchpwdom  9492