Theorem difprsn2 4686

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 4620	. . 3 ⊢ {𝐴, 𝐵} = {𝐵, 𝐴}
2	1	difeq1i 4007	. 2 ⊢ ({𝐴, 𝐵} ∖ {𝐵}) = ({𝐵, 𝐴} ∖ {𝐵})
3		necom 2987	. . 3 ⊢ (𝐴 ≠ 𝐵 ↔ 𝐵 ≠ 𝐴)
4		difprsn1 4685	. . 3 ⊢ (𝐵 ≠ 𝐴 → ({𝐵, 𝐴} ∖ {𝐵}) = {𝐴})
5	3, 4	sylbi 220	. 2 ⊢ (𝐴 ≠ 𝐵 → ({𝐵, 𝐴} ∖ {𝐵}) = {𝐴})
6	2, 5	syl5eq 2785	1 ⊢ (𝐴 ≠ 𝐵 → ({𝐴, 𝐵} ∖ {𝐵}) = {𝐴})

Syntax hints:   → wi 4   = wceq 1542   ≠ wne 2934   ∖ cdif 3838  {csn 4513  {cpr 4515

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 1916  ax-6 1974  ax-7 2019  ax-8 2115  ax-9 2123  ax-ext 2710

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-tru 1545  df-fal 1555  df-ex 1787  df-sb 2074  df-clab 2717  df-cleq 2730  df-clel 2811  df-ne 2935  df-ral 3058  df-rab 3062  df-v 3399  df-dif 3844  df-un 3846  df-in 3848  df-nul 4210  df-sn 4514  df-pr 4516

This theorem is referenced by:  f12dfv  7035  pmtrprfval  18726  nbgr2vtx1edg  27284  nbuhgr2vtx1edgb  27286  nfrgr2v  28201  cycpm2tr  30955  ldepsnlinc  45367