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Theorem xpsneng 6832
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 4610 . . 3  |-  ( x  =  A  ->  (
x  X.  { y } )  =  ( A  X.  { y } ) )
2 id 21 . . 3  |-  ( x  =  A  ->  x  =  A )
31, 2breq12d 3933 . 2  |-  ( x  =  A  ->  (
( x  X.  {
y } )  ~~  x 
<->  ( A  X.  {
y } )  ~~  A ) )
4 sneq 3555 . . . 4  |-  ( y  =  B  ->  { y }  =  { B } )
54xpeq2d 4620 . . 3  |-  ( y  =  B  ->  ( A  X.  { y } )  =  ( A  X.  { B }
) )
65breq1d 3930 . 2  |-  ( y  =  B  ->  (
( A  X.  {
y } )  ~~  A 
<->  ( A  X.  { B } )  ~~  A
) )
7 vex 2730 . . 3  |-  x  e. 
_V
8 vex 2730 . . 3  |-  y  e. 
_V
97, 8xpsnen 6831 . 2  |-  ( x  X.  { y } )  ~~  x
103, 6, 9vtocl2g 2785 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 3544   class class class wbr 3920    X. cxp 4578    ~~ cen 6746
This theorem is referenced by:  xp1en  6833  xpsnen2g  6840  xpdom3  6845  disjen  6903  unxpdom2  6956  sucxpdom  6957  uncdadom  7681  cdaun  7682  cdaen  7683  cda1dif  7686  cdacomen  7691  cdaassen  7692  xpcdaen  7693  mapcdaen  7694  cdaxpdom  7699  cdafi  7700  cdainf  7702  infcda1  7703  pwcdadom  7726  isfin4-3  7825  pwcdandom  8169  gchxpidm  8171
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 1926  ax-ext 2234  ax-sep 4038  ax-nul 4046  ax-pow 4082  ax-pr 4108  ax-un 4403
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 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-ral 2513  df-rex 2514  df-rab 2516  df-v 2729  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-nul 3363  df-if 3471  df-pw 3532  df-sn 3550  df-pr 3551  df-op 3553  df-uni 3728  df-int 3761  df-br 3921  df-opab 3975  df-mpt 3976  df-id 4202  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-fun 4602  df-fn 4603  df-f 4604  df-f1 4605  df-fo 4606  df-f1o 4607  df-en 6750
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