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Theorem xpsnen2g 6689
Description: A set is equinumerous to its Cartesian product with a singleton on the left. (Contributed by Stefan O'Rear, 21-Nov-2014.)
Assertion
Ref Expression
xpsnen2g  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( { A }  X.  B )  ~~  B
)

Proof of Theorem xpsnen2g
StepHypRef Expression
1 snexg 4076 . . 3  |-  ( A  e.  V  ->  { A }  e.  _V )
2 xpcomeng 6688 . . 3  |-  ( ( { A }  e.  _V  /\  B  e.  W
)  ->  ( { A }  X.  B
)  ~~  ( B  X.  { A } ) )
31, 2sylan 279 . 2  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( { A }  X.  B )  ~~  ( B  X.  { A }
) )
4 xpsneng 6682 . . 3  |-  ( ( B  e.  W  /\  A  e.  V )  ->  ( B  X.  { A } )  ~~  B
)
54ancoms 266 . 2  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( B  X.  { A } )  ~~  B
)
6 entr 6644 . 2  |-  ( ( ( { A }  X.  B )  ~~  ( B  X.  { A }
)  /\  ( B  X.  { A } ) 
~~  B )  -> 
( { A }  X.  B )  ~~  B
)
73, 5, 6syl2anc 406 1  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( { A }  X.  B )  ~~  B
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    e. wcel 1463   _Vcvv 2658   {csn 3495   class class class wbr 3897    X. cxp 4505    ~~ cen 6598
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 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4014  ax-pow 4066  ax-pr 4099  ax-un 4323
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ral 2396  df-rex 2397  df-v 2660  df-sbc 2881  df-un 3043  df-in 3045  df-ss 3052  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-int 3740  df-br 3898  df-opab 3958  df-mpt 3959  df-id 4183  df-xp 4513  df-rel 4514  df-cnv 4515  df-co 4516  df-dm 4517  df-rn 4518  df-res 4519  df-ima 4520  df-iota 5056  df-fun 5093  df-fn 5094  df-f 5095  df-f1 5096  df-fo 5097  df-f1o 5098  df-fv 5099  df-1st 6004  df-2nd 6005  df-er 6395  df-en 6601
This theorem is referenced by:  djucomen  7036  djuassen  7037  xpdjuen  7038
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