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Theorem difprsn1 4800
Description: Removal of a singleton from an unordered pair. (Contributed by Thierry Arnoux, 4-Feb-2017.)
Assertion
Ref Expression
difprsn1 (𝐴𝐵 → ({𝐴, 𝐵} ∖ {𝐴}) = {𝐵})

Proof of Theorem difprsn1
StepHypRef Expression
1 necom 2994 . 2 (𝐵𝐴𝐴𝐵)
2 df-pr 4629 . . . . . 6 {𝐴, 𝐵} = ({𝐴} ∪ {𝐵})
32equncomi 4160 . . . . 5 {𝐴, 𝐵} = ({𝐵} ∪ {𝐴})
43difeq1i 4122 . . . 4 ({𝐴, 𝐵} ∖ {𝐴}) = (({𝐵} ∪ {𝐴}) ∖ {𝐴})
5 difun2 4481 . . . 4 (({𝐵} ∪ {𝐴}) ∖ {𝐴}) = ({𝐵} ∖ {𝐴})
64, 5eqtri 2765 . . 3 ({𝐴, 𝐵} ∖ {𝐴}) = ({𝐵} ∖ {𝐴})
7 disjsn2 4712 . . . 4 (𝐵𝐴 → ({𝐵} ∩ {𝐴}) = ∅)
8 disj3 4454 . . . 4 (({𝐵} ∩ {𝐴}) = ∅ ↔ {𝐵} = ({𝐵} ∖ {𝐴}))
97, 8sylib 218 . . 3 (𝐵𝐴 → {𝐵} = ({𝐵} ∖ {𝐴}))
106, 9eqtr4id 2796 . 2 (𝐵𝐴 → ({𝐴, 𝐵} ∖ {𝐴}) = {𝐵})
111, 10sylbir 235 1 (𝐴𝐵 → ({𝐴, 𝐵} ∖ {𝐴}) = {𝐵})
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1540  wne 2940  cdif 3948  cun 3949  cin 3950  c0 4333  {csn 4626  {cpr 4628
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3062  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-nul 4334  df-sn 4627  df-pr 4629
This theorem is referenced by:  difprsn2  4801  f12dfv  7293  pmtrprfval  19505  nbgr2vtx1edg  29367  nbuhgr2vtx1edgb  29369  nfrgr2v  30291  mptprop  32707  cycpm2tr  33139  eulerpartlemgf  34381  coinflippvt  34487  ldepsnlinc  48425
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