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Theorem difprsn1 4362
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 2876 . 2 (𝐵𝐴𝐴𝐵)
2 disjsn2 4279 . . . 4 (𝐵𝐴 → ({𝐵} ∩ {𝐴}) = ∅)
3 disj3 4054 . . . 4 (({𝐵} ∩ {𝐴}) = ∅ ↔ {𝐵} = ({𝐵} ∖ {𝐴}))
42, 3sylib 208 . . 3 (𝐵𝐴 → {𝐵} = ({𝐵} ∖ {𝐴}))
5 df-pr 4213 . . . . . 6 {𝐴, 𝐵} = ({𝐴} ∪ {𝐵})
65equncomi 3792 . . . . 5 {𝐴, 𝐵} = ({𝐵} ∪ {𝐴})
76difeq1i 3757 . . . 4 ({𝐴, 𝐵} ∖ {𝐴}) = (({𝐵} ∪ {𝐴}) ∖ {𝐴})
8 difun2 4081 . . . 4 (({𝐵} ∪ {𝐴}) ∖ {𝐴}) = ({𝐵} ∖ {𝐴})
97, 8eqtri 2673 . . 3 ({𝐴, 𝐵} ∖ {𝐴}) = ({𝐵} ∖ {𝐴})
104, 9syl6reqr 2704 . 2 (𝐵𝐴 → ({𝐴, 𝐵} ∖ {𝐴}) = {𝐵})
111, 10sylbir 225 1 (𝐴𝐵 → ({𝐴, 𝐵} ∖ {𝐴}) = {𝐵})
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1523  wne 2823  cdif 3604  cun 3605  cin 3606  c0 3948  {csn 4210  {cpr 4212
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rab 2950  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-sn 4211  df-pr 4213
This theorem is referenced by:  difprsn2  4363  f12dfv  6569  pmtrprfval  17953  nbgr2vtx1edg  26291  nbuhgr2vtx1edgb  26293  nfrgr2v  27252  eulerpartlemgf  30569  coinflippvt  30674  ldepsnlinc  42622
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