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Theorem difprsn1 4732
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 3074 . 2 (𝐵𝐴𝐴𝐵)
2 disjsn2 4647 . . . 4 (𝐵𝐴 → ({𝐵} ∩ {𝐴}) = ∅)
3 disj3 4406 . . . 4 (({𝐵} ∩ {𝐴}) = ∅ ↔ {𝐵} = ({𝐵} ∖ {𝐴}))
42, 3sylib 219 . . 3 (𝐵𝐴 → {𝐵} = ({𝐵} ∖ {𝐴}))
5 df-pr 4567 . . . . . 6 {𝐴, 𝐵} = ({𝐴} ∪ {𝐵})
65equncomi 4135 . . . . 5 {𝐴, 𝐵} = ({𝐵} ∪ {𝐴})
76difeq1i 4099 . . . 4 ({𝐴, 𝐵} ∖ {𝐴}) = (({𝐵} ∪ {𝐴}) ∖ {𝐴})
8 difun2 4432 . . . 4 (({𝐵} ∪ {𝐴}) ∖ {𝐴}) = ({𝐵} ∖ {𝐴})
97, 8eqtri 2849 . . 3 ({𝐴, 𝐵} ∖ {𝐴}) = ({𝐵} ∖ {𝐴})
104, 9syl6reqr 2880 . 2 (𝐵𝐴 → ({𝐴, 𝐵} ∖ {𝐴}) = {𝐵})
111, 10sylbir 236 1 (𝐴𝐵 → ({𝐴, 𝐵} ∖ {𝐴}) = {𝐵})
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1530  wne 3021  cdif 3937  cun 3938  cin 3939  c0 4295  {csn 4564  {cpr 4566
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-ral 3148  df-rab 3152  df-v 3502  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-nul 4296  df-sn 4565  df-pr 4567
This theorem is referenced by:  difprsn2  4733  f12dfv  7026  pmtrprfval  18551  nbgr2vtx1edg  27065  nbuhgr2vtx1edgb  27067  nfrgr2v  27984  mptprop  30366  cycpm2tr  30694  eulerpartlemgf  31542  coinflippvt  31647  ldepsnlinc  44465
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