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Theorem xpcomeng 6922
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
StepHypRef Expression
1 xpeq1 4691 . . 3  |-  ( x  =  A  ->  (
x  X.  y )  =  ( A  X.  y ) )
2 xpeq2 4692 . . 3  |-  ( x  =  A  ->  (
y  X.  x )  =  ( y  X.  A ) )
31, 2breq12d 4010 . 2  |-  ( x  =  A  ->  (
( x  X.  y
)  ~~  ( y  X.  x )  <->  ( A  X.  y )  ~~  (
y  X.  A ) ) )
4 xpeq2 4692 . . 3  |-  ( y  =  B  ->  ( A  X.  y )  =  ( A  X.  B
) )
5 xpeq1 4691 . . 3  |-  ( y  =  B  ->  (
y  X.  A )  =  ( B  X.  A ) )
64, 5breq12d 4010 . 2  |-  ( y  =  B  ->  (
( A  X.  y
)  ~~  ( y  X.  A )  <->  ( A  X.  B )  ~~  ( B  X.  A ) ) )
7 vex 2766 . . 3  |-  x  e. 
_V
8 vex 2766 . . 3  |-  y  e. 
_V
97, 8xpcomen 6921 . 2  |-  ( x  X.  y )  ~~  ( y  X.  x
)
103, 6, 9vtocl2g 2822 1  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( A  X.  B
)  ~~  ( B  X.  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 6    /\ wa 360    = wceq 1619    e. wcel 1621   class class class wbr 3997    X. cxp 4659    ~~ cen 6828
This theorem is referenced by:  xpsnen2g  6923  xpdom1g  6927  omxpen  6932  xpfir  7053  infxp  7809  infmap2  7812
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2239  ax-sep 4115  ax-nul 4123  ax-pow 4160  ax-pr 4186  ax-un 4484
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2122  df-mo 2123  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-ne 2423  df-ral 2523  df-rex 2524  df-rab 2527  df-v 2765  df-sbc 2967  df-dif 3130  df-un 3132  df-in 3134  df-ss 3141  df-nul 3431  df-if 3540  df-pw 3601  df-sn 3620  df-pr 3621  df-op 3623  df-uni 3802  df-br 3998  df-opab 4052  df-mpt 4053  df-id 4281  df-xp 4675  df-rel 4676  df-cnv 4677  df-co 4678  df-dm 4679  df-rn 4680  df-res 4681  df-ima 4682  df-fun 4683  df-fn 4684  df-f 4685  df-f1 4686  df-fo 4687  df-f1o 4688  df-fv 4689  df-1st 6056  df-2nd 6057  df-en 6832
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