Theorem difprsn1 4733

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 3069	. 2 ⊢ (𝐵 ≠ 𝐴 ↔ 𝐴 ≠ 𝐵)
2		disjsn2 4648	. . . 4 ⊢ (𝐵 ≠ 𝐴 → ({𝐵} ∩ {𝐴}) = ∅)
3		disj3 4403	. . . 4 ⊢ (({𝐵} ∩ {𝐴}) = ∅ ↔ {𝐵} = ({𝐵} ∖ {𝐴}))
4	2, 3	sylib 220	. . 3 ⊢ (𝐵 ≠ 𝐴 → {𝐵} = ({𝐵} ∖ {𝐴}))
5		df-pr 4570	. . . . . 6 ⊢ {𝐴, 𝐵} = ({𝐴} ∪ {𝐵})
6	5	equncomi 4131	. . . . 5 ⊢ {𝐴, 𝐵} = ({𝐵} ∪ {𝐴})
7	6	difeq1i 4095	. . . 4 ⊢ ({𝐴, 𝐵} ∖ {𝐴}) = (({𝐵} ∪ {𝐴}) ∖ {𝐴})
8		difun2 4429	. . . 4 ⊢ (({𝐵} ∪ {𝐴}) ∖ {𝐴}) = ({𝐵} ∖ {𝐴})
9	7, 8	eqtri 2844	. . 3 ⊢ ({𝐴, 𝐵} ∖ {𝐴}) = ({𝐵} ∖ {𝐴})
10	4, 9	syl6reqr 2875	. 2 ⊢ (𝐵 ≠ 𝐴 → ({𝐴, 𝐵} ∖ {𝐴}) = {𝐵})
11	1, 10	sylbir 237	1 ⊢ (𝐴 ≠ 𝐵 → ({𝐴, 𝐵} ∖ {𝐴}) = {𝐵})

Syntax hints:   → wi 4   = wceq 1537   ≠ wne 3016   ∖ cdif 3933   ∪ cun 3934   ∩ cin 3935  ∅c0 4291  {csn 4567  {cpr 4569

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rab 3147  df-v 3496  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-sn 4568  df-pr 4570

This theorem is referenced by:  difprsn2  4734  f12dfv  7030  pmtrprfval  18615  nbgr2vtx1edg  27132  nbuhgr2vtx1edgb  27134  nfrgr2v  28051  mptprop  30434  cycpm2tr  30761  eulerpartlemgf  31637  coinflippvt  31742  ldepsnlinc  44583