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Theorem xpcomeng 6690
Description: Commutative law for equinumerosity of Cartesian product. Proposition 4.22(d) of [Mendelson] p. 254. (Contributed by NM, 27-Mar-2006.)
Assertion
Ref Expression
xpcomeng  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( A  X.  B
)  ~~  ( B  X.  A ) )

Proof of Theorem xpcomeng
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 xpeq1 4523 . . 3  |-  ( x  =  A  ->  (
x  X.  y )  =  ( A  X.  y ) )
2 xpeq2 4524 . . 3  |-  ( x  =  A  ->  (
y  X.  x )  =  ( y  X.  A ) )
31, 2breq12d 3912 . 2  |-  ( x  =  A  ->  (
( x  X.  y
)  ~~  ( y  X.  x )  <->  ( A  X.  y )  ~~  (
y  X.  A ) ) )
4 xpeq2 4524 . . 3  |-  ( y  =  B  ->  ( A  X.  y )  =  ( A  X.  B
) )
5 xpeq1 4523 . . 3  |-  ( y  =  B  ->  (
y  X.  A )  =  ( B  X.  A ) )
64, 5breq12d 3912 . 2  |-  ( y  =  B  ->  (
( A  X.  y
)  ~~  ( y  X.  A )  <->  ( A  X.  B )  ~~  ( B  X.  A ) ) )
7 vex 2663 . . 3  |-  x  e. 
_V
8 vex 2663 . . 3  |-  y  e. 
_V
97, 8xpcomen 6689 . 2  |-  ( x  X.  y )  ~~  ( y  X.  x
)
103, 6, 9vtocl2g 2724 1  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( A  X.  B
)  ~~  ( B  X.  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1316    e. wcel 1465   class class class wbr 3899    X. cxp 4507    ~~ cen 6600
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101  ax-un 4325
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-v 2662  df-sbc 2883  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-br 3900  df-opab 3960  df-mpt 3961  df-id 4185  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-iota 5058  df-fun 5095  df-fn 5096  df-f 5097  df-f1 5098  df-fo 5099  df-f1o 5100  df-fv 5101  df-1st 6006  df-2nd 6007  df-en 6603
This theorem is referenced by:  xpsnen2g  6691  xpdom1g  6695  hashxp  10540
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