Theorem xpsnen2g 8053

Description: A set is equinumerous to its Cartesian product with a singleton on the left. (Contributed by Stefan O'Rear, 21-Nov-2014.)

Assertion

Ref	Expression
xpsnen2g	|- ( ( A e. V /\ B e. W ) -> ( { A } X. B ) ~~ B )

Proof of Theorem xpsnen2g

Step	Hyp	Ref	Expression
1		snex 4908	. . . 4 |- { A } e. _V
2		xpcomeng 8052	. . . 4 |- ( ( { A } e. _V /\ B e. W ) -> ( { A } X. B ) ~~ ( B X. { A } ) )
3	1, 2	mpan 706	. . 3 |- ( B e. W -> ( { A } X. B ) ~~ ( B X. { A } ) )
4	3	adantl 482	. 2 |- ( ( A e. V /\ B e. W ) -> ( { A } X. B ) ~~ ( B X. { A } ) )
5		xpsneng 8045	. . 3 |- ( ( B e. W /\ A e. V ) -> ( B X. { A } ) ~~ B )
6	5	ancoms 469	. 2 |- ( ( A e. V /\ B e. W ) -> ( B X. { A } ) ~~ B )
7		entr 8008	. 2 |- ( ( ( { A } X. B ) ~~ ( B X. { A } ) /\ ( B X. { A } ) ~~ B ) -> ( { A } X. B ) ~~ B )
8	4, 6, 7	syl2anc 693	1 |- ( ( A e. V /\ B e. W ) -> ( { A } X. B ) ~~ B )

Syntax hints:   -> wi 4   /\ wa 384   e. wcel 1990   _Vcvv 3200   {csn 4177   class class class wbr 4653   X. cxp 5112   ~~ 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-int 4476  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-1st 7168  df-2nd 7169  df-er 7742  df-en 7956

This theorem is referenced by:  unxpwdom2  8493  ackbij1lem8  9049  lgsquadlem1  25105  lgsquadlem2  25106