ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  xpcomen Unicode version

Theorem xpcomen 6727
Description: Commutative law for equinumerosity of Cartesian 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
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 xpcomen.1 . . 3  |-  A  e. 
_V
2 xpcomen.2 . . 3  |-  B  e. 
_V
31, 2xpex 4660 . 2  |-  ( A  X.  B )  e. 
_V
42, 1xpex 4660 . 2  |-  ( B  X.  A )  e. 
_V
5 eqid 2140 . . 3  |-  ( x  e.  ( A  X.  B )  |->  U. `' { x } )  =  ( x  e.  ( A  X.  B
)  |->  U. `' { x } )
65xpcomf1o 6725 . 2  |-  ( x  e.  ( A  X.  B )  |->  U. `' { x } ) : ( A  X.  B ) -1-1-onto-> ( B  X.  A
)
7 f1oen2g 6655 . 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 1316 1  |-  ( A  X.  B )  ~~  ( B  X.  A
)
Colors of variables: wff set class
Syntax hints:    e. wcel 1481   _Vcvv 2689   {csn 3530   U.cuni 3742   class class class wbr 3935    |-> cmpt 3995    X. cxp 4543   `'ccnv 4544   -1-1-onto->wf1o 5128    ~~ cen 6638
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 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4052  ax-pow 4104  ax-pr 4137  ax-un 4361
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2691  df-sbc 2913  df-un 3078  df-in 3080  df-ss 3087  df-pw 3515  df-sn 3536  df-pr 3537  df-op 3539  df-uni 3743  df-br 3936  df-opab 3996  df-mpt 3997  df-id 4221  df-xp 4551  df-rel 4552  df-cnv 4553  df-co 4554  df-dm 4555  df-rn 4556  df-iota 5094  df-fun 5131  df-fn 5132  df-f 5133  df-f1 5134  df-fo 5135  df-f1o 5136  df-fv 5137  df-1st 6044  df-2nd 6045  df-en 6641
This theorem is referenced by:  xpcomeng  6728
  Copyright terms: Public domain W3C validator