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Theorem difprsn1 4299
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 2843 . 2 (𝐵𝐴𝐴𝐵)
2 disjsn2 4217 . . . 4 (𝐵𝐴 → ({𝐵} ∩ {𝐴}) = ∅)
3 disj3 3993 . . . 4 (({𝐵} ∩ {𝐴}) = ∅ ↔ {𝐵} = ({𝐵} ∖ {𝐴}))
42, 3sylib 208 . . 3 (𝐵𝐴 → {𝐵} = ({𝐵} ∖ {𝐴}))
5 df-pr 4151 . . . . . 6 {𝐴, 𝐵} = ({𝐴} ∪ {𝐵})
65equncomi 3737 . . . . 5 {𝐴, 𝐵} = ({𝐵} ∪ {𝐴})
76difeq1i 3702 . . . 4 ({𝐴, 𝐵} ∖ {𝐴}) = (({𝐵} ∪ {𝐴}) ∖ {𝐴})
8 difun2 4020 . . . 4 (({𝐵} ∪ {𝐴}) ∖ {𝐴}) = ({𝐵} ∖ {𝐴})
97, 8eqtri 2643 . . 3 ({𝐴, 𝐵} ∖ {𝐴}) = ({𝐵} ∖ {𝐴})
104, 9syl6reqr 2674 . 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 1480  wne 2790  cdif 3552  cun 3553  cin 3554  c0 3891  {csn 4148  {cpr 4150
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rab 2916  df-v 3188  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-sn 4149  df-pr 4151
This theorem is referenced by:  difprsn2  4300  f12dfv  6483  pmtrprfval  17828  nbgr2vtx1edg  26133  nbuhgr2vtx1edgb  26135  nfrgr2v  27000  eulerpartlemgf  30222  coinflippvt  30327  ldepsnlinc  41585
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