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Theorem difprsn2 4153
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 4094 . . 3  |-  { A ,  B }  =  { B ,  A }
21difeq1i 3604 . 2  |-  ( { A ,  B }  \  { B } )  =  ( { B ,  A }  \  { B } )
3 necom 2723 . . 3  |-  ( A  =/=  B  <->  B  =/=  A )
4 difprsn1 4152 . . 3  |-  ( B  =/=  A  ->  ( { B ,  A }  \  { B } )  =  { A }
)
53, 4sylbi 195 . 2  |-  ( A  =/=  B  ->  ( { B ,  A }  \  { B } )  =  { A }
)
62, 5syl5eq 2507 1  |-  ( A  =/=  B  ->  ( { A ,  B }  \  { B } )  =  { A }
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1398    =/= wne 2649    \ cdif 3458   {csn 4016   {cpr 4018
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rab 2813  df-v 3108  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-sn 4017  df-pr 4019
This theorem is referenced by:  f12dfv  6154  pmtrprfval  16711  cusgra2v  24664  frgra2v  25201  ldepsnlinc  33363
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