Theorem difprsn2 4741

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 4671	. . 3 ⊢ {𝐴, 𝐵} = {𝐵, 𝐴}
2	1	difeq1i 4060	. 2 ⊢ ({𝐴, 𝐵} ∖ {𝐵}) = ({𝐵, 𝐴} ∖ {𝐵})
3		necom 2988	. . 3 ⊢ (𝐴 ≠ 𝐵 ↔ 𝐵 ≠ 𝐴)
4		difprsn1 4740	. . 3 ⊢ (𝐵 ≠ 𝐴 → ({𝐵, 𝐴} ∖ {𝐵}) = {𝐴})
5	3, 4	sylbi 218	. 2 ⊢ (𝐴 ≠ 𝐵 → ({𝐵, 𝐴} ∖ {𝐵}) = {𝐴})
6	2, 5	eqtrid 2787	1 ⊢ (𝐴 ≠ 𝐵 → ({𝐴, 𝐵} ∖ {𝐵}) = {𝐴})

Syntax hints:   → wi 4   = wceq 1547   ≠ wne 2935   ∖ cdif 3887  {csn 4562  {cpr 4564

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2712

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-tru 1550  df-fal 1560  df-ex 1787  df-sb 2074  df-clab 2719  df-cleq 2732  df-clel 2815  df-ne 2936  df-ral 3055  df-rab 3393  df-v 3434  df-dif 3893  df-un 3895  df-in 3897  df-nul 4269  df-sn 4563  df-pr 4565

This theorem is referenced by:  f12dfv  7224  pmtrprfval  19460  nbgr2vtx1edg  29444  nbuhgr2vtx1edgb  29446  nfrgr2v  30367  indsupp  32953  cycpm2tr  33207  drngmxidl  33567  ldepsnlinc  49006