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Theorem xpcomeng 7163
Description: Commutative law for equinumerosity of cross 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 4855 . . 3  |-  ( x  =  A  ->  (
x  X.  y )  =  ( A  X.  y ) )
2 xpeq2 4856 . . 3  |-  ( x  =  A  ->  (
y  X.  x )  =  ( y  X.  A ) )
31, 2breq12d 4189 . 2  |-  ( x  =  A  ->  (
( x  X.  y
)  ~~  ( y  X.  x )  <->  ( A  X.  y )  ~~  (
y  X.  A ) ) )
4 xpeq2 4856 . . 3  |-  ( y  =  B  ->  ( A  X.  y )  =  ( A  X.  B
) )
5 xpeq1 4855 . . 3  |-  ( y  =  B  ->  (
y  X.  A )  =  ( B  X.  A ) )
64, 5breq12d 4189 . 2  |-  ( y  =  B  ->  (
( A  X.  y
)  ~~  ( y  X.  A )  <->  ( A  X.  B )  ~~  ( B  X.  A ) ) )
7 vex 2923 . . 3  |-  x  e. 
_V
8 vex 2923 . . 3  |-  y  e. 
_V
97, 8xpcomen 7162 . 2  |-  ( x  X.  y )  ~~  ( y  X.  x
)
103, 6, 9vtocl2g 2979 1  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( A  X.  B
)  ~~  ( B  X.  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1721   class class class wbr 4176    X. cxp 4839    ~~ cen 7069
This theorem is referenced by:  xpsnen2g  7164  xpdom1g  7168  omxpen  7173  xpfir  7294  infxp  8055  infmap2  8058
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389  ax-sep 4294  ax-nul 4302  ax-pow 4341  ax-pr 4367  ax-un 4664
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-ral 2675  df-rex 2676  df-rab 2679  df-v 2922  df-sbc 3126  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-nul 3593  df-if 3704  df-pw 3765  df-sn 3784  df-pr 3785  df-op 3787  df-uni 3980  df-br 4177  df-opab 4231  df-mpt 4232  df-id 4462  df-xp 4847  df-rel 4848  df-cnv 4849  df-co 4850  df-dm 4851  df-rn 4852  df-iota 5381  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-1st 6312  df-2nd 6313  df-en 7073
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