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Theorem xpsneng 6880
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
StepHypRef Expression
1 xpeq1 4656 . . 3  |-  ( x  =  A  ->  (
x  X.  { y } )  =  ( A  X.  { y } ) )
2 id 21 . . 3  |-  ( x  =  A  ->  x  =  A )
31, 2breq12d 3976 . 2  |-  ( x  =  A  ->  (
( x  X.  {
y } )  ~~  x 
<->  ( A  X.  {
y } )  ~~  A ) )
4 sneq 3592 . . . 4  |-  ( y  =  B  ->  { y }  =  { B } )
54xpeq2d 4666 . . 3  |-  ( y  =  B  ->  ( A  X.  { y } )  =  ( A  X.  { B }
) )
65breq1d 3973 . 2  |-  ( y  =  B  ->  (
( A  X.  {
y } )  ~~  A 
<->  ( A  X.  { B } )  ~~  A
) )
7 vex 2743 . . 3  |-  x  e. 
_V
8 vex 2743 . . 3  |-  y  e. 
_V
97, 8xpsnen 6879 . 2  |-  ( x  X.  { y } )  ~~  x
103, 6, 9vtocl2g 2798 1  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( A  X.  { B } )  ~~  A
)
Colors of variables: wff set class
Syntax hints:    -> wi 6    /\ wa 360    = wceq 1619    e. wcel 1621   {csn 3581   class class class wbr 3963    X. cxp 4624    ~~ cen 6793
This theorem is referenced by:  xp1en  6881  xpsnen2g  6888  xpdom3  6893  disjen  6951  unxpdom2  7004  sucxpdom  7005  uncdadom  7730  cdaun  7731  cdaen  7732  cda1dif  7735  cdacomen  7740  cdaassen  7741  xpcdaen  7742  mapcdaen  7743  cdaxpdom  7748  cdafi  7749  cdainf  7751  infcda1  7752  pwcdadom  7775  isfin4-3  7874  pwcdandom  8222  gchxpidm  8224
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 2237  ax-sep 4081  ax-nul 4089  ax-pow 4126  ax-pr 4152  ax-un 4449
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 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-ral 2520  df-rex 2521  df-rab 2523  df-v 2742  df-dif 3097  df-un 3099  df-in 3101  df-ss 3108  df-nul 3398  df-if 3507  df-pw 3568  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3769  df-int 3804  df-br 3964  df-opab 4018  df-mpt 4019  df-id 4246  df-xp 4640  df-rel 4641  df-cnv 4642  df-co 4643  df-dm 4644  df-rn 4645  df-fun 4648  df-fn 4649  df-f 4650  df-f1 4651  df-fo 4652  df-f1o 4653  df-en 6797
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