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Theorem difprsn2 4805
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 4737 . . 3 {𝐴, 𝐵} = {𝐵, 𝐴}
21difeq1i 4119 . 2 ({𝐴, 𝐵} ∖ {𝐵}) = ({𝐵, 𝐴} ∖ {𝐵})
3 necom 2995 . . 3 (𝐴𝐵𝐵𝐴)
4 difprsn1 4804 . . 3 (𝐵𝐴 → ({𝐵, 𝐴} ∖ {𝐵}) = {𝐴})
53, 4sylbi 216 . 2 (𝐴𝐵 → ({𝐵, 𝐴} ∖ {𝐵}) = {𝐴})
62, 5eqtrid 2785 1 (𝐴𝐵 → ({𝐴, 𝐵} ∖ {𝐵}) = {𝐴})
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1542  wne 2941  cdif 3946  {csn 4629  {cpr 4631
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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2942  df-ral 3063  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-nul 4324  df-sn 4630  df-pr 4632
This theorem is referenced by:  f12dfv  7271  pmtrprfval  19355  nbgr2vtx1edg  28607  nbuhgr2vtx1edgb  28609  nfrgr2v  29525  cycpm2tr  32278  drngmxidl  32593  ldepsnlinc  47189
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