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Theorem xpsneng 6709
Description: A set is equinumerous to its Cartesian product with a singleton. Proposition 4.22(c) of [Mendelson] p. 254. (Contributed by NM, 22-Oct-2004.)
Assertion
Ref Expression
xpsneng  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( A  X.  { B } )  ~~  A
)

Proof of Theorem xpsneng
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 xpeq1 4548 . . 3  |-  ( x  =  A  ->  (
x  X.  { y } )  =  ( A  X.  { y } ) )
2 id 19 . . 3  |-  ( x  =  A  ->  x  =  A )
31, 2breq12d 3937 . 2  |-  ( x  =  A  ->  (
( x  X.  {
y } )  ~~  x 
<->  ( A  X.  {
y } )  ~~  A ) )
4 sneq 3533 . . . 4  |-  ( y  =  B  ->  { y }  =  { B } )
54xpeq2d 4558 . . 3  |-  ( y  =  B  ->  ( A  X.  { y } )  =  ( A  X.  { B }
) )
65breq1d 3934 . 2  |-  ( y  =  B  ->  (
( A  X.  {
y } )  ~~  A 
<->  ( A  X.  { B } )  ~~  A
) )
7 vex 2684 . . 3  |-  x  e. 
_V
8 vex 2684 . . 3  |-  y  e. 
_V
97, 8xpsnen 6708 . 2  |-  ( x  X.  { y } )  ~~  x
103, 6, 9vtocl2g 2745 1  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( A  X.  { B } )  ~~  A
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1331    e. wcel 1480   {csn 3522   class class class wbr 3924    X. cxp 4532    ~~ cen 6625
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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-pow 4093  ax-pr 4126  ax-un 4350
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-v 2683  df-un 3070  df-in 3072  df-ss 3079  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-int 3767  df-br 3925  df-opab 3985  df-mpt 3986  df-id 4210  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-rn 4545  df-fun 5120  df-fn 5121  df-f 5122  df-f1 5123  df-fo 5124  df-f1o 5125  df-en 6628
This theorem is referenced by:  xp1en  6710  xpsnen2g  6716  xpdom3m  6721  hashxp  10565  pwf1oexmid  13183
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