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Theorem difprsn2 4734
Description: Removal of a singleton from an unordered pair. (Contributed by Alexander van der Vekens, 5-Oct-2017.)
Assertion
Ref Expression
difprsn2 (𝐴𝐵 → ({𝐴, 𝐵} ∖ {𝐵}) = {𝐴})

Proof of Theorem difprsn2
StepHypRef Expression
1 prcom 4668 . . 3 {𝐴, 𝐵} = {𝐵, 𝐴}
21difeq1i 4053 . 2 ({𝐴, 𝐵} ∖ {𝐵}) = ({𝐵, 𝐴} ∖ {𝐵})
3 necom 2997 . . 3 (𝐴𝐵𝐵𝐴)
4 difprsn1 4733 . . 3 (𝐵𝐴 → ({𝐵, 𝐴} ∖ {𝐵}) = {𝐴})
53, 4sylbi 216 . 2 (𝐴𝐵 → ({𝐵, 𝐴} ∖ {𝐵}) = {𝐴})
62, 5eqtrid 2790 1 (𝐴𝐵 → ({𝐴, 𝐵} ∖ {𝐵}) = {𝐴})
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1539  wne 2943  cdif 3884  {csn 4561  {cpr 4563
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ne 2944  df-ral 3069  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-nul 4257  df-sn 4562  df-pr 4564
This theorem is referenced by:  f12dfv  7145  pmtrprfval  19095  nbgr2vtx1edg  27717  nbuhgr2vtx1edgb  27719  nfrgr2v  28636  cycpm2tr  31386  ldepsnlinc  45849
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