Theorem difprsn1 3847

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 2597	. 2 ⊢ (B ≠ A ↔ A ≠ B)
2		disjsn2 3787	. . . 4 ⊢ (B ≠ A → ({B} ∩ {A}) = ∅)
3		disj3 3595	. . . 4 ⊢ (({B} ∩ {A}) = ∅ ↔ {B} = ({B} ∖ {A}))
4	2, 3	sylib 188	. . 3 ⊢ (B ≠ A → {B} = ({B} ∖ {A}))
5		df-pr 3742	. . . . . 6 ⊢ {A, B} = ({A} ∪ {B})
6	5	equncomi 3410	. . . . 5 ⊢ {A, B} = ({B} ∪ {A})
7	6	difeq1i 3381	. . . 4 ⊢ ({A, B} ∖ {A}) = (({B} ∪ {A}) ∖ {A})
8		difun2 3629	. . . 4 ⊢ (({B} ∪ {A}) ∖ {A}) = ({B} ∖ {A})
9	7, 8	eqtri 2373	. . 3 ⊢ ({A, B} ∖ {A}) = ({B} ∖ {A})
10	4, 9	syl6reqr 2404	. 2 ⊢ (B ≠ A → ({A, B} ∖ {A}) = {B})
11	1, 10	sylbir 204	1 ⊢ (A ≠ B → ({A, B} ∖ {A}) = {B})

Syntax hints:   → wi 4   = wceq 1642   ≠ wne 2516   ∖ cdif 3206   ∪ cun 3207   ∩ cin 3208  ∅c0 3550  {csn 3737  {cpr 3738

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ne 2518  df-ral 2619  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-ss 3259  df-nul 3551  df-sn 3741  df-pr 3742

This theorem is referenced by:  difprsn2  3848