Theorem difprsn1 4769

Description: Removal of a singleton from an unordered pair. (Contributed by Thierry Arnoux, 4-Feb-2017.)

Assertion

Ref	Expression
difprsn1	⊢ (𝐴 ≠ 𝐵 → ({𝐴, 𝐵} ∖ {𝐴}) = {𝐵})

Proof of Theorem difprsn1

Step	Hyp	Ref	Expression
1		necom 3011	. 2 ⊢ (𝐵 ≠ 𝐴 ↔ 𝐴 ≠ 𝐵)
2		df-pr 4593	. . . . . 6 ⊢ {𝐴, 𝐵} = ({𝐴} ∪ {𝐵})
3	2	equncomi 4115	. . . . 5 ⊢ {𝐴, 𝐵} = ({𝐵} ∪ {𝐴})
4	3	difeq1i 4078	. . . 4 ⊢ ({𝐴, 𝐵} ∖ {𝐴}) = (({𝐵} ∪ {𝐴}) ∖ {𝐴})
5		difun2 4443	. . . 4 ⊢ (({𝐵} ∪ {𝐴}) ∖ {𝐴}) = ({𝐵} ∖ {𝐴})
6	4, 5	eqtri 2786	. . 3 ⊢ ({𝐴, 𝐵} ∖ {𝐴}) = ({𝐵} ∖ {𝐴})
7		disjsn2 4679	. . . 4 ⊢ (𝐵 ≠ 𝐴 → ({𝐵} ∩ {𝐴}) = ∅)
8		disj3 4415	. . . 4 ⊢ (({𝐵} ∩ {𝐴}) = ∅ ↔ {𝐵} = ({𝐵} ∖ {𝐴}))
9	7, 8	sylib 221	. . 3 ⊢ (𝐵 ≠ 𝐴 → {𝐵} = ({𝐵} ∖ {𝐴}))
10	6, 9	eqtr4id 2817	. 2 ⊢ (𝐵 ≠ 𝐴 → ({𝐴, 𝐵} ∖ {𝐴}) = {𝐵})
11	1, 10	sylbir 238	1 ⊢ (𝐴 ≠ 𝐵 → ({𝐴, 𝐵} ∖ {𝐴}) = {𝐵})

Syntax hints:   → wi 4   = wceq 1570   ≠ wne 2958   ∖ cdif 3903   ∪ cun 3904   ∩ cin 3905  ∅c0 4287  {csn 4590  {cpr 4592

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-tru 1573  df-fal 1583  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-ne 2959  df-ral 3080  df-rab 3417  df-v 3457  df-dif 3909  df-un 3911  df-in 3913  df-nul 4288  df-sn 4591  df-pr 4593

This theorem is referenced by:  difprsn2  4770  f12dfv  7273  pmtrprfval  19558  nbgr2vtx1edg  29678  nbuhgr2vtx1edgb  29680  nfrgr2v  30601  mptprop  33021  indfsid  33167  cycpm2tr  33417  eulerpartlemgf  34747  coinflippvt  34853  ldepsnlinc  49265