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Theorem difprsn2 4272
 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 4211 . . 3 {𝐴, 𝐵} = {𝐵, 𝐴}
21difeq1i 3686 . 2 ({𝐴, 𝐵} ∖ {𝐵}) = ({𝐵, 𝐴} ∖ {𝐵})
3 necom 2835 . . 3 (𝐴𝐵𝐵𝐴)
4 difprsn1 4271 . . 3 (𝐵𝐴 → ({𝐵, 𝐴} ∖ {𝐵}) = {𝐴})
53, 4sylbi 206 . 2 (𝐴𝐵 → ({𝐵, 𝐴} ∖ {𝐵}) = {𝐴})
62, 5syl5eq 2656 1 (𝐴𝐵 → ({𝐴, 𝐵} ∖ {𝐵}) = {𝐴})
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1475   ≠ wne 2780   ∖ cdif 3537  {csn 4125  {cpr 4127 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-sn 4126  df-pr 4128 This theorem is referenced by:  f12dfv  6429  pmtrprfval  17730  cusgra2v  25991  frgra2v  26526  nbgr2vtx1edg  40572  nbuhgr2vtx1edgb  40574  nfrgr2v  41442  ldepsnlinc  42091
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