Theorem enpw 6088

Description: If A and B are equinumerous, then so are their power sets. Theorem XI.1.36 of [Rosser] p. 369. (Contributed by SF, 17-Mar-2015.)

Assertion

Ref	Expression
enpw	|- (A ~~ B -> ~PA ~~ ~PB)

Proof of Theorem enpw

Dummy variables a b are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		brex 4690	. 2 |- (A ~~ B -> (A e. _V /\ B e. _V))
2		breq1 4643	. . . 4 |- (a = A -> (a ~~ b <-> A ~~ b))
3		pweq 3726	. . . . 5 |- (a = A -> ~Pa = ~PA)
4	3	breq1d 4650	. . . 4 |- (a = A -> (~Pa ~~ ~Pb <-> ~PA ~~ ~Pb))
5	2, 4	imbi12d 311	. . 3 |- (a = A -> ((a ~~ b -> ~Pa ~~ ~Pb) <-> (A ~~ b -> ~PA ~~ ~Pb)))
6		breq2 4644	. . . 4 |- (b = B -> (A ~~ b <-> A ~~ B))
7		pweq 3726	. . . . 5 |- (b = B -> ~Pb = ~PB)
8	7	breq2d 4652	. . . 4 |- (b = B -> (~PA ~~ ~Pb <-> ~PA ~~ ~PB))
9	6, 8	imbi12d 311	. . 3 |- (b = B -> ((A ~~ b -> ~PA ~~ ~Pb) <-> (A ~~ B -> ~PA ~~ ~PB)))
10		enmap2 6069	. . . 4 |- (a ~~ b -> ({_V, (/)} ^m a) ~~ ({_V, (/)} ^m b))
11		vn0 3558	. . . . . . 7 |- _V =/= (/)
12		eqid 2353	. . . . . . 7 |- {_V, (/)} = {_V, (/)}
13		vvex 4110	. . . . . . . 8 |- _V e. _V
14		0ex 4111	. . . . . . . 8 |- (/) e. _V
15		vex 2863	. . . . . . . 8 |- a e. _V
16	13, 14, 15	enprmapc 6084	. . . . . . 7 |- ((_V =/= (/) /\ {_V, (/)} = {_V, (/)}) -> ({_V, (/)} ^m a) ~~ ~Pa)
17	11, 12, 16	mp2an 653	. . . . . 6 |- ({_V, (/)} ^m a) ~~ ~Pa
18		ensym 6038	. . . . . 6 |- (~Pa ~~ ({_V, (/)} ^m a) <-> ({_V, (/)} ^m a) ~~ ~Pa)
19	17, 18	mpbir 200	. . . . 5 |- ~Pa ~~ ({_V, (/)} ^m a)
20		vex 2863	. . . . . . . 8 |- b e. _V
21	13, 14, 20	enprmapc 6084	. . . . . . 7 |- ((_V =/= (/) /\ {_V, (/)} = {_V, (/)}) -> ({_V, (/)} ^m b) ~~ ~Pb)
22	11, 12, 21	mp2an 653	. . . . . 6 |- ({_V, (/)} ^m b) ~~ ~Pb
23		entr 6039	. . . . . 6 |- ((({_V, (/)} ^m a) ~~ ({_V, (/)} ^m b) /\ ({_V, (/)} ^m b) ~~ ~Pb) -> ({_V, (/)} ^m a) ~~ ~Pb)
24	22, 23	mpan2 652	. . . . 5 |- (({_V, (/)} ^m a) ~~ ({_V, (/)} ^m b) -> ({_V, (/)} ^m a) ~~ ~Pb)
25		entr 6039	. . . . 5 |- ((~Pa ~~ ({_V, (/)} ^m a) /\ ({_V, (/)} ^m a) ~~ ~Pb) -> ~Pa ~~ ~Pb)
26	19, 24, 25	sylancr 644	. . . 4 |- (({_V, (/)} ^m a) ~~ ({_V, (/)} ^m b) -> ~Pa ~~ ~Pb)
27	10, 26	syl 15	. . 3 |- (a ~~ b -> ~Pa ~~ ~Pb)
28	5, 9, 27	vtocl2g 2919	. 2 |- ((A e. _V /\ B e. _V) -> (A ~~ B -> ~PA ~~ ~PB))
29	1, 28	mpcom 32	1 |- (A ~~ B -> ~PA ~~ ~PB)

Syntax hints:   -> wi 4   /\ wa 358   = wceq 1642   e. wcel 1710   =/= wne 2517  _Vcvv 2860  (/)c0 3551  ~Pcpw 3723  {cpr 3739   class class class wbr 4640  (class class class)co 5526   ^m cmap 6000   ~~ cen 6029

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4079  ax-xp 4080  ax-cnv 4081  ax-1c 4082  ax-sset 4083  ax-si 4084  ax-ins2 4085  ax-ins3 4086  ax-typlower 4087  ax-sn 4088

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ne 2519  df-ral 2620  df-rex 2621  df-reu 2622  df-rmo 2623  df-rab 2624  df-v 2862  df-sbc 3048  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-symdif 3217  df-ss 3260  df-pss 3262  df-nul 3552  df-if 3664  df-pw 3725  df-sn 3742  df-pr 3743  df-uni 3893  df-int 3928  df-opk 4059  df-1c 4137  df-pw1 4138  df-uni1 4139  df-xpk 4186  df-cnvk 4187  df-ins2k 4188  df-ins3k 4189  df-imak 4190  df-cok 4191  df-p6 4192  df-sik 4193  df-ssetk 4194  df-imagek 4195  df-idk 4196  df-iota 4340  df-0c 4378  df-addc 4379  df-nnc 4380  df-fin 4381  df-lefin 4441  df-ltfin 4442  df-ncfin 4443  df-tfin 4444  df-evenfin 4445  df-oddfin 4446  df-sfin 4447  df-spfin 4448  df-phi 4566  df-op 4567  df-proj1 4568  df-proj2 4569  df-opab 4624  df-br 4641  df-1st 4724  df-swap 4725  df-sset 4726  df-co 4727  df-ima 4728  df-si 4729  df-id 4768  df-xp 4785  df-cnv 4786  df-rn 4787  df-dm 4788  df-res 4789  df-fun 4790  df-fn 4791  df-f 4792  df-f1 4793  df-fo 4794  df-f1o 4795  df-fv 4796  df-2nd 4798  df-ov 5527  df-oprab 5529  df-mpt 5653  df-mpt2 5655  df-txp 5737  df-compose 5749  df-ins2 5751  df-ins3 5753  df-image 5755  df-ins4 5757  df-si3 5759  df-funs 5761  df-map 6002  df-en 6030

This theorem is referenced by:  ce2le  6234