Theorem difprsn1 4330

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 2847	. 2 ⊢ (𝐵 ≠ 𝐴 ↔ 𝐴 ≠ 𝐵)
2		disjsn2 4247	. . . 4 ⊢ (𝐵 ≠ 𝐴 → ({𝐵} ∩ {𝐴}) = ∅)
3		disj3 4021	. . . 4 ⊢ (({𝐵} ∩ {𝐴}) = ∅ ↔ {𝐵} = ({𝐵} ∖ {𝐴}))
4	2, 3	sylib 208	. . 3 ⊢ (𝐵 ≠ 𝐴 → {𝐵} = ({𝐵} ∖ {𝐴}))
5		df-pr 4180	. . . . . 6 ⊢ {𝐴, 𝐵} = ({𝐴} ∪ {𝐵})
6	5	equncomi 3759	. . . . 5 ⊢ {𝐴, 𝐵} = ({𝐵} ∪ {𝐴})
7	6	difeq1i 3724	. . . 4 ⊢ ({𝐴, 𝐵} ∖ {𝐴}) = (({𝐵} ∪ {𝐴}) ∖ {𝐴})
8		difun2 4048	. . . 4 ⊢ (({𝐵} ∪ {𝐴}) ∖ {𝐴}) = ({𝐵} ∖ {𝐴})
9	7, 8	eqtri 2644	. . 3 ⊢ ({𝐴, 𝐵} ∖ {𝐴}) = ({𝐵} ∖ {𝐴})
10	4, 9	syl6reqr 2675	. 2 ⊢ (𝐵 ≠ 𝐴 → ({𝐴, 𝐵} ∖ {𝐴}) = {𝐵})
11	1, 10	sylbir 225	1 ⊢ (𝐴 ≠ 𝐵 → ({𝐴, 𝐵} ∖ {𝐴}) = {𝐵})

Syntax hints:   → wi 4   = wceq 1483   ≠ wne 2794   ∖ cdif 3571   ∪ cun 3572   ∩ cin 3573  ∅c0 3915  {csn 4177  {cpr 4179

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-sn 4178  df-pr 4180

This theorem is referenced by:  difprsn2  4331  f12dfv  6529  pmtrprfval  17907  nbgr2vtx1edg  26246  nbuhgr2vtx1edgb  26248  nfrgr2v  27136  eulerpartlemgf  30441  coinflippvt  30546  ldepsnlinc  42297