Theorem difprsn1 4730

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 2996	. 2 ⊢ (𝐵 ≠ 𝐴 ↔ 𝐴 ≠ 𝐵)
2		df-pr 4561	. . . . . 6 ⊢ {𝐴, 𝐵} = ({𝐴} ∪ {𝐵})
3	2	equncomi 4085	. . . . 5 ⊢ {𝐴, 𝐵} = ({𝐵} ∪ {𝐴})
4	3	difeq1i 4049	. . . 4 ⊢ ({𝐴, 𝐵} ∖ {𝐴}) = (({𝐵} ∪ {𝐴}) ∖ {𝐴})
5		difun2 4411	. . . 4 ⊢ (({𝐵} ∪ {𝐴}) ∖ {𝐴}) = ({𝐵} ∖ {𝐴})
6	4, 5	eqtri 2766	. . 3 ⊢ ({𝐴, 𝐵} ∖ {𝐴}) = ({𝐵} ∖ {𝐴})
7		disjsn2 4645	. . . 4 ⊢ (𝐵 ≠ 𝐴 → ({𝐵} ∩ {𝐴}) = ∅)
8		disj3 4384	. . . 4 ⊢ (({𝐵} ∩ {𝐴}) = ∅ ↔ {𝐵} = ({𝐵} ∖ {𝐴}))
9	7, 8	sylib 217	. . 3 ⊢ (𝐵 ≠ 𝐴 → {𝐵} = ({𝐵} ∖ {𝐴}))
10	6, 9	eqtr4id 2798	. 2 ⊢ (𝐵 ≠ 𝐴 → ({𝐴, 𝐵} ∖ {𝐴}) = {𝐵})
11	1, 10	sylbir 234	1 ⊢ (𝐴 ≠ 𝐵 → ({𝐴, 𝐵} ∖ {𝐴}) = {𝐵})

Syntax hints:   → wi 4   = wceq 1539   ≠ wne 2942   ∖ cdif 3880   ∪ cun 3881   ∩ cin 3882  ∅c0 4253  {csn 4558  {cpr 4560

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-tru 1542  df-fal 1552  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-ne 2943  df-ral 3068  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-nul 4254  df-sn 4559  df-pr 4561

This theorem is referenced by:  difprsn2  4731  f12dfv  7126  pmtrprfval  19010  nbgr2vtx1edg  27620  nbuhgr2vtx1edgb  27622  nfrgr2v  28537  mptprop  30933  cycpm2tr  31288  eulerpartlemgf  32246  coinflippvt  32351  ldepsnlinc  45737