Theorem difprsn1 4758

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 2986	. 2 ⊢ (𝐵 ≠ 𝐴 ↔ 𝐴 ≠ 𝐵)
2		df-pr 4585	. . . . . 6 ⊢ {𝐴, 𝐵} = ({𝐴} ∪ {𝐵})
3	2	equncomi 4114	. . . . 5 ⊢ {𝐴, 𝐵} = ({𝐵} ∪ {𝐴})
4	3	difeq1i 4076	. . . 4 ⊢ ({𝐴, 𝐵} ∖ {𝐴}) = (({𝐵} ∪ {𝐴}) ∖ {𝐴})
5		difun2 4435	. . . 4 ⊢ (({𝐵} ∪ {𝐴}) ∖ {𝐴}) = ({𝐵} ∖ {𝐴})
6	4, 5	eqtri 2760	. . 3 ⊢ ({𝐴, 𝐵} ∖ {𝐴}) = ({𝐵} ∖ {𝐴})
7		disjsn2 4671	. . . 4 ⊢ (𝐵 ≠ 𝐴 → ({𝐵} ∩ {𝐴}) = ∅)
8		disj3 4408	. . . 4 ⊢ (({𝐵} ∩ {𝐴}) = ∅ ↔ {𝐵} = ({𝐵} ∖ {𝐴}))
9	7, 8	sylib 218	. . 3 ⊢ (𝐵 ≠ 𝐴 → {𝐵} = ({𝐵} ∖ {𝐴}))
10	6, 9	eqtr4id 2791	. 2 ⊢ (𝐵 ≠ 𝐴 → ({𝐴, 𝐵} ∖ {𝐴}) = {𝐵})
11	1, 10	sylbir 235	1 ⊢ (𝐴 ≠ 𝐵 → ({𝐴, 𝐵} ∖ {𝐴}) = {𝐵})

Syntax hints:   → wi 4   = wceq 1542   ≠ wne 2933   ∖ cdif 3900   ∪ cun 3901   ∩ cin 3902  ∅c0 4287  {csn 4582  {cpr 4584

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ne 2934  df-ral 3053  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-in 3910  df-nul 4288  df-sn 4583  df-pr 4585

This theorem is referenced by:  difprsn2  4759  f12dfv  7229  pmtrprfval  19428  nbgr2vtx1edg  29435  nbuhgr2vtx1edgb  29437  nfrgr2v  30359  mptprop  32787  indfsid  32961  cycpm2tr  33212  eulerpartlemgf  34556  coinflippvt  34662  ldepsnlinc  48862