Theorem xpcomen 8051

Description: Commutative law for equinumerosity of Cartesian product. Proposition 4.22(d) of [Mendelson] p. 254. (Contributed by NM, 5-Jan-2004.) (Revised by Mario Carneiro, 15-Nov-2014.)

Hypotheses

Ref	Expression
xpcomen.1	|- A e. _V
xpcomen.2	|- B e. _V

Assertion

Ref	Expression
xpcomen	|- ( A X. B ) ~~ ( B X. A )

Proof of Theorem xpcomen

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		xpcomen.1	. . 3 |- A e. _V
2		xpcomen.2	. . 3 |- B e. _V
3	1, 2	xpex 6962	. 2 |- ( A X. B ) e. _V
4	2, 1	xpex 6962	. 2 |- ( B X. A ) e. _V
5		eqid 2622	. . 3 |- ( x e. ( A X. B ) |-> U. `' { x } ) = ( x e. ( A X. B ) |-> U. `' { x } )
6	5	xpcomf1o 8049	. 2 |- ( x e. ( A X. B ) |-> U. `' { x } ) : ( A X. B ) -1-1-onto-> ( B X. A )
7		f1oen2g 7972	. 2 |- ( ( ( A X. B ) e. _V /\ ( B X. A ) e. _V /\ ( x e. ( A X. B ) |-> U. `' { x } ) : ( A X. B ) -1-1-onto-> ( B X. A ) ) -> ( A X. B ) ~~ ( B X. A ) )
8	3, 4, 6, 7	mp3an 1424	1 |- ( A X. B ) ~~ ( B X. A )

Syntax hints:   e. wcel 1990   _Vcvv 3200   {csn 4177   U.cuni 4436   class class class wbr 4653   |-> cmpt 4729   X. cxp 5112   `'ccnv 5113   -1-1-onto->wf1o 5887   ~~ cen 7952

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-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-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-1st 7168  df-2nd 7169  df-en 7956

This theorem is referenced by:  xpcomeng  8052  hashxplem  13220