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Theorem xpsnen2g 6774
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 4145 . . 3  |-  ( A  e.  V  ->  { A }  e.  _V )
2 xpcomeng 6773 . . 3  |-  ( ( { A }  e.  _V  /\  B  e.  W
)  ->  ( { A }  X.  B
)  ~~  ( B  X.  { A } ) )
31, 2sylan 281 . 2  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( { A }  X.  B )  ~~  ( B  X.  { A }
) )
4 xpsneng 6767 . . 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 6729 . 2  |-  ( ( ( { A }  X.  B )  ~~  ( B  X.  { A }
)  /\  ( B  X.  { A } ) 
~~  B )  -> 
( { A }  X.  B )  ~~  B
)
73, 5, 6syl2anc 409 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 2128   _Vcvv 2712   {csn 3560   class class class wbr 3965    X. cxp 4584    ~~ cen 6683
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 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-sep 4082  ax-pow 4135  ax-pr 4169  ax-un 4393
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-v 2714  df-sbc 2938  df-un 3106  df-in 3108  df-ss 3115  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-int 3808  df-br 3966  df-opab 4026  df-mpt 4027  df-id 4253  df-xp 4592  df-rel 4593  df-cnv 4594  df-co 4595  df-dm 4596  df-rn 4597  df-res 4598  df-ima 4599  df-iota 5135  df-fun 5172  df-fn 5173  df-f 5174  df-f1 5175  df-fo 5176  df-f1o 5177  df-fv 5178  df-1st 6088  df-2nd 6089  df-er 6480  df-en 6686
This theorem is referenced by:  djucomen  7151  djuassen  7152  xpdjuen  7153
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