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Theorem xpsnen2g 6334
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 3964 . . 3  |-  ( A  e.  V  ->  { A }  e.  _V )
2 xpcomeng 6333 . . 3  |-  ( ( { A }  e.  _V  /\  B  e.  W
)  ->  ( { A }  X.  B
)  ~~  ( B  X.  { A } ) )
31, 2sylan 271 . 2  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( { A }  X.  B )  ~~  ( B  X.  { A }
) )
4 xpsneng 6327 . . 3  |-  ( ( B  e.  W  /\  A  e.  V )  ->  ( B  X.  { A } )  ~~  B
)
54ancoms 259 . 2  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( B  X.  { A } )  ~~  B
)
6 entr 6295 . 2  |-  ( ( ( { A }  X.  B )  ~~  ( B  X.  { A }
)  /\  ( B  X.  { A } ) 
~~  B )  -> 
( { A }  X.  B )  ~~  B
)
73, 5, 6syl2anc 397 1  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( { A }  X.  B )  ~~  B
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 101    e. wcel 1409   _Vcvv 2574   {csn 3403   class class class wbr 3792    X. cxp 4371    ~~ cen 6250
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3903  ax-pow 3955  ax-pr 3972  ax-un 4198
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-sbc 2788  df-un 2950  df-in 2952  df-ss 2959  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-uni 3609  df-int 3644  df-br 3793  df-opab 3847  df-mpt 3848  df-id 4058  df-xp 4379  df-rel 4380  df-cnv 4381  df-co 4382  df-dm 4383  df-rn 4384  df-res 4385  df-ima 4386  df-iota 4895  df-fun 4932  df-fn 4933  df-f 4934  df-f1 4935  df-fo 4936  df-f1o 4937  df-fv 4938  df-1st 5795  df-2nd 5796  df-er 6137  df-en 6253
This theorem is referenced by: (None)
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