Theorem difprsn2 4768

Description: Removal of a singleton from an unordered pair. (Contributed by Alexander van der Vekens, 5-Oct-2017.)

Assertion

Ref	Expression
difprsn2	⊢ (𝐴 ≠ 𝐵 → ({𝐴, 𝐵} ∖ {𝐵}) = {𝐴})

Proof of Theorem difprsn2

Step	Hyp	Ref	Expression
1		prcom 4697	. . 3 ⊢ {𝐴, 𝐵} = {𝐵, 𝐴}
2	1	difeq1i 4076	. 2 ⊢ ({𝐴, 𝐵} ∖ {𝐵}) = ({𝐵, 𝐴} ∖ {𝐵})
3		necom 3010	. . 3 ⊢ (𝐴 ≠ 𝐵 ↔ 𝐵 ≠ 𝐴)
4		difprsn1 4767	. . 3 ⊢ (𝐵 ≠ 𝐴 → ({𝐵, 𝐴} ∖ {𝐵}) = {𝐴})
5	3, 4	sylbi 220	. 2 ⊢ (𝐴 ≠ 𝐵 → ({𝐵, 𝐴} ∖ {𝐵}) = {𝐴})
6	2, 5	eqtrid 2809	1 ⊢ (𝐴 ≠ 𝐵 → ({𝐴, 𝐵} ∖ {𝐵}) = {𝐴})

This proof depends on syntax axioms:   → wi 4   = wceq 1569   ≠ wne 2957   ∖ cdif 3901  {csn 4588  {cpr 4590

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-ext 2734

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-tru 1572  df-fal 1582  df-ex 1809  df-sb 2096  df-clab 2741  df-cleq 2754  df-clel 2837  df-ne 2958  df-ral 3079  df-rab 3416  df-v 3456  df-dif 3907  df-un 3909  df-in 3911  df-nul 4286  df-sn 4589  df-pr 4591

This theorem is used by:  f12dfv  7271  pmtrprfval  19563  nbgr2vtx1edg  29711  nbuhgr2vtx1edgb  29713  nfrgr2v  30634  indsupp  33198  cycpm2tr  33448  drngmxidl  33768  ldepsnlinc  49316