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Theorem difprsn1 4805
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 4634 . . . . . 6 {𝐴, 𝐵} = ({𝐴} ∪ {𝐵})
32equncomi 4170 . . . . 5 {𝐴, 𝐵} = ({𝐵} ∪ {𝐴})
43difeq1i 4132 . . . 4 ({𝐴, 𝐵} ∖ {𝐴}) = (({𝐵} ∪ {𝐴}) ∖ {𝐴})
5 difun2 4487 . . . 4 (({𝐵} ∪ {𝐴}) ∖ {𝐴}) = ({𝐵} ∖ {𝐴})
64, 5eqtri 2763 . . 3 ({𝐴, 𝐵} ∖ {𝐴}) = ({𝐵} ∖ {𝐴})
7 disjsn2 4717 . . . 4 (𝐵𝐴 → ({𝐵} ∩ {𝐴}) = ∅)
8 disj3 4460 . . . 4 (({𝐵} ∩ {𝐴}) = ∅ ↔ {𝐵} = ({𝐵} ∖ {𝐴}))
97, 8sylib 218 . . 3 (𝐵𝐴 → {𝐵} = ({𝐵} ∖ {𝐴}))
106, 9eqtr4id 2794 . 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 1537  wne 2938  cdif 3960  cun 3961  cin 3962  c0 4339  {csn 4631  {cpr 4633
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-ral 3060  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-nul 4340  df-sn 4632  df-pr 4634
This theorem is referenced by:  difprsn2  4806  f12dfv  7293  pmtrprfval  19520  nbgr2vtx1edg  29382  nbuhgr2vtx1edgb  29384  nfrgr2v  30301  mptprop  32713  cycpm2tr  33122  eulerpartlemgf  34361  coinflippvt  34466  ldepsnlinc  48354
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