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Theorem xpsnen2g 7600
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
StepHypRef Expression
1 snex 4681 . . . 4  |-  { A }  e.  _V
2 xpcomeng 7599 . . . 4  |-  ( ( { A }  e.  _V  /\  B  e.  W
)  ->  ( { A }  X.  B
)  ~~  ( B  X.  { A } ) )
31, 2mpan 670 . . 3  |-  ( B  e.  W  ->  ( { A }  X.  B
)  ~~  ( B  X.  { A } ) )
43adantl 466 . 2  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( { A }  X.  B )  ~~  ( B  X.  { A }
) )
5 xpsneng 7592 . . 3  |-  ( ( B  e.  W  /\  A  e.  V )  ->  ( B  X.  { A } )  ~~  B
)
65ancoms 453 . 2  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( B  X.  { A } )  ~~  B
)
7 entr 7557 . 2  |-  ( ( ( { A }  X.  B )  ~~  ( B  X.  { A }
)  /\  ( B  X.  { A } ) 
~~  B )  -> 
( { A }  X.  B )  ~~  B
)
84, 6, 7syl2anc 661 1  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( { A }  X.  B )  ~~  B
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    e. wcel 1762   _Vcvv 3106   {csn 4020   class class class wbr 4440    X. cxp 4990    ~~ cen 7503
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3108  df-sbc 3325  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-int 4276  df-br 4441  df-opab 4499  df-mpt 4500  df-id 4788  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-1st 6774  df-2nd 6775  df-er 7301  df-en 7507
This theorem is referenced by:  unxpwdom2  8003  ackbij1lem8  8596  lgsquadlem1  23350  lgsquadlem2  23351
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