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Theorem difprsn2 4726
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 4660 . . 3 {𝐴, 𝐵} = {𝐵, 𝐴}
21difeq1i 4092 . 2 ({𝐴, 𝐵} ∖ {𝐵}) = ({𝐵, 𝐴} ∖ {𝐵})
3 necom 3066 . . 3 (𝐴𝐵𝐵𝐴)
4 difprsn1 4725 . . 3 (𝐵𝐴 → ({𝐵, 𝐴} ∖ {𝐵}) = {𝐴})
53, 4sylbi 218 . 2 (𝐴𝐵 → ({𝐵, 𝐴} ∖ {𝐵}) = {𝐴})
62, 5syl5eq 2865 1 (𝐴𝐵 → ({𝐴, 𝐵} ∖ {𝐵}) = {𝐴})
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1528  wne 3013  cdif 3930  {csn 4557  {cpr 4559
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rab 3144  df-v 3494  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-sn 4558  df-pr 4560
This theorem is referenced by:  f12dfv  7021  pmtrprfval  18544  nbgr2vtx1edg  27059  nbuhgr2vtx1edgb  27061  nfrgr2v  27978  cycpm2tr  30688  ldepsnlinc  44491
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