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Theorem difprsn1 4802
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 2992 . 2 (𝐵𝐴𝐴𝐵)
2 df-pr 4630 . . . . . 6 {𝐴, 𝐵} = ({𝐴} ∪ {𝐵})
32equncomi 4154 . . . . 5 {𝐴, 𝐵} = ({𝐵} ∪ {𝐴})
43difeq1i 4117 . . . 4 ({𝐴, 𝐵} ∖ {𝐴}) = (({𝐵} ∪ {𝐴}) ∖ {𝐴})
5 difun2 4479 . . . 4 (({𝐵} ∪ {𝐴}) ∖ {𝐴}) = ({𝐵} ∖ {𝐴})
64, 5eqtri 2758 . . 3 ({𝐴, 𝐵} ∖ {𝐴}) = ({𝐵} ∖ {𝐴})
7 disjsn2 4715 . . . 4 (𝐵𝐴 → ({𝐵} ∩ {𝐴}) = ∅)
8 disj3 4452 . . . 4 (({𝐵} ∩ {𝐴}) = ∅ ↔ {𝐵} = ({𝐵} ∖ {𝐴}))
97, 8sylib 217 . . 3 (𝐵𝐴 → {𝐵} = ({𝐵} ∖ {𝐴}))
106, 9eqtr4id 2789 . 2 (𝐵𝐴 → ({𝐴, 𝐵} ∖ {𝐴}) = {𝐵})
111, 10sylbir 234 1 (𝐴𝐵 → ({𝐴, 𝐵} ∖ {𝐴}) = {𝐵})
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1539  wne 2938  cdif 3944  cun 3945  cin 3946  c0 4321  {csn 4627  {cpr 4629
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 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-ext 2701
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-tru 1542  df-fal 1552  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2722  df-clel 2808  df-ne 2939  df-ral 3060  df-rab 3431  df-v 3474  df-dif 3950  df-un 3952  df-in 3954  df-nul 4322  df-sn 4628  df-pr 4630
This theorem is referenced by:  difprsn2  4803  f12dfv  7273  pmtrprfval  19396  nbgr2vtx1edg  28874  nbuhgr2vtx1edgb  28876  nfrgr2v  29792  mptprop  32187  cycpm2tr  32548  eulerpartlemgf  33676  coinflippvt  33781  ldepsnlinc  47276
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