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Theorem xpcomen 6921
Description: Commutative law for equinumerosity of cross product. Proposition 4.22(d) of [Mendelson] p. 254. (Contributed by NM, 5-Jan-2004.) (Revised by Mario Carneiro, 15-Nov-2014.)
Hypotheses
Ref Expression
xpcomen.1  |-  A  e. 
_V
xpcomen.2  |-  B  e. 
_V
Assertion
Ref Expression
xpcomen  |-  ( A  X.  B )  ~~  ( B  X.  A
)

Proof of Theorem xpcomen
StepHypRef Expression
1 xpcomen.1 . . 3  |-  A  e. 
_V
2 xpcomen.2 . . 3  |-  B  e. 
_V
31, 2xpex 4789 . 2  |-  ( A  X.  B )  e. 
_V
42, 1xpex 4789 . 2  |-  ( B  X.  A )  e. 
_V
5 eqid 2258 . . 3  |-  ( x  e.  ( A  X.  B )  |->  U. `' { x } )  =  ( x  e.  ( A  X.  B
)  |->  U. `' { x } )
65xpcomf1o 6919 . 2  |-  ( x  e.  ( A  X.  B )  |->  U. `' { x } ) : ( A  X.  B ) -1-1-onto-> ( B  X.  A
)
7 f1oen2g 6846 . 2  |-  ( ( ( A  X.  B
)  e.  _V  /\  ( B  X.  A
)  e.  _V  /\  ( x  e.  ( A  X.  B )  |->  U. `' { x } ) : ( A  X.  B ) -1-1-onto-> ( B  X.  A
) )  ->  ( A  X.  B )  ~~  ( B  X.  A
) )
83, 4, 6, 7mp3an 1282 1  |-  ( A  X.  B )  ~~  ( B  X.  A
)
Colors of variables: wff set class
Syntax hints:    e. wcel 1621   _Vcvv 2763   {csn 3614   U.cuni 3801   class class class wbr 3997    e. cmpt 4051    X. cxp 4659   `'ccnv 4660   -1-1-onto->wf1o 4672    ~~ cen 6828
This theorem is referenced by:  xpcomeng  6922  hashxplem  11350
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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