Theorem difprsn2 3848

Description: Removal of a singleton from an unordered pair. (Contributed by Alexander van der Vekens, 5-Oct-2017.)

Assertion

Ref	Expression
difprsn2	⊢ (A ≠ B → ({A, B} ∖ {B}) = {A})

Proof of Theorem difprsn2

Step	Hyp	Ref	Expression
1		prcom 3798	. . 3 ⊢ {A, B} = {B, A}
2	1	difeq1i 3381	. 2 ⊢ ({A, B} ∖ {B}) = ({B, A} ∖ {B})
3		necom 2597	. . 3 ⊢ (A ≠ B ↔ B ≠ A)
4		difprsn1 3847	. . 3 ⊢ (B ≠ A → ({B, A} ∖ {B}) = {A})
5	3, 4	sylbi 187	. 2 ⊢ (A ≠ B → ({B, A} ∖ {B}) = {A})
6	2, 5	syl5eq 2397	1 ⊢ (A ≠ B → ({A, B} ∖ {B}) = {A})

Syntax hints:   → wi 4   = wceq 1642   ≠ wne 2516   ∖ cdif 3206  {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: (None)