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Theorem xpsneng 7156
Description: A set is equinumerous to its cross-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 4855 . . 3  |-  ( x  =  A  ->  (
x  X.  { y } )  =  ( A  X.  { y } ) )
2 id 20 . . 3  |-  ( x  =  A  ->  x  =  A )
31, 2breq12d 4189 . 2  |-  ( x  =  A  ->  (
( x  X.  {
y } )  ~~  x 
<->  ( A  X.  {
y } )  ~~  A ) )
4 sneq 3789 . . . 4  |-  ( y  =  B  ->  { y }  =  { B } )
54xpeq2d 4865 . . 3  |-  ( y  =  B  ->  ( A  X.  { y } )  =  ( A  X.  { B }
) )
65breq1d 4186 . 2  |-  ( y  =  B  ->  (
( A  X.  {
y } )  ~~  A 
<->  ( A  X.  { B } )  ~~  A
) )
7 vex 2923 . . 3  |-  x  e. 
_V
8 vex 2923 . . 3  |-  y  e. 
_V
97, 8xpsnen 7155 . 2  |-  ( x  X.  { y } )  ~~  x
103, 6, 9vtocl2g 2979 1  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( A  X.  { B } )  ~~  A
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1721   {csn 3778   class class class wbr 4176    X. cxp 4839    ~~ cen 7069
This theorem is referenced by:  xp1en  7157  xpsnen2g  7164  xpdom3  7169  disjen  7227  unxpdom2  7280  sucxpdom  7281  uncdadom  8011  cdaun  8012  cdaen  8013  cda1dif  8016  cdacomen  8021  cdaassen  8022  xpcdaen  8023  mapcdaen  8024  cdaxpdom  8029  cdafi  8030  cdainf  8032  infcda1  8033  pwcdadom  8056  isfin4-3  8155  pwcdandom  8502  gchxpidm  8504
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389  ax-sep 4294  ax-nul 4302  ax-pow 4341  ax-pr 4367  ax-un 4664
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-ral 2675  df-rex 2676  df-rab 2679  df-v 2922  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-nul 3593  df-if 3704  df-pw 3765  df-sn 3784  df-pr 3785  df-op 3787  df-uni 3980  df-int 4015  df-br 4177  df-opab 4231  df-mpt 4232  df-id 4462  df-xp 4847  df-rel 4848  df-cnv 4849  df-co 4850  df-dm 4851  df-rn 4852  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-en 7073
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