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Theorem difprsn2 3750
Description: Removal of a singleton from an unordered pair. (Contributed by Alexander van der Vekens, 5-Oct-2017.)
Assertion
Ref Expression
difprsn2  |-  ( A  =/=  B  ->  ( { A ,  B }  \  { B } )  =  { A }
)

Proof of Theorem difprsn2
StepHypRef Expression
1 prcom 3686 . . 3  |-  { A ,  B }  =  { B ,  A }
21difeq1i 3264 . 2  |-  ( { A ,  B }  \  { B } )  =  ( { B ,  A }  \  { B } )
3 necom 2444 . . 3  |-  ( A  =/=  B  <->  B  =/=  A )
4 difprsn1 3749 . . 3  |-  ( B  =/=  A  ->  ( { B ,  A }  \  { B } )  =  { A }
)
53, 4sylbi 121 . 2  |-  ( A  =/=  B  ->  ( { B ,  A }  \  { B } )  =  { A }
)
62, 5eqtrid 2234 1  |-  ( A  =/=  B  ->  ( { A ,  B }  \  { B } )  =  { A }
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1364    =/= wne 2360    \ cdif 3141   {csn 3610   {cpr 3611
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171
This theorem depends on definitions:  df-bi 117  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-ral 2473  df-rab 2477  df-v 2754  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-sn 3616  df-pr 3617
This theorem is referenced by: (None)
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