Theorem disjsn2 3787

Description: Intersection of distinct singletons is disjoint. (Contributed by NM, 25-May-1998.)

Assertion

Ref	Expression
disjsn2	⊢ (A ≠ B → ({A} ∩ {B}) = ∅)

Proof of Theorem disjsn2

Step	Hyp	Ref	Expression
1		elsni 3757	. . . 4 ⊢ (B ∈ {A} → B = A)
2	1	eqcomd 2358	. . 3 ⊢ (B ∈ {A} → A = B)
3	2	necon3ai 2556	. 2 ⊢ (A ≠ B → ¬ B ∈ {A})
4		disjsn 3786	. 2 ⊢ (({A} ∩ {B}) = ∅ ↔ ¬ B ∈ {A})
5	3, 4	sylibr 203	1 ⊢ (A ≠ B → ({A} ∩ {B}) = ∅)

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1642   ∈ wcel 1710   ≠ wne 2516   ∩ cin 3208  ∅c0 3550  {csn 3737

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-dif 3215  df-nul 3551  df-sn 3741

This theorem is referenced by:  difprsn1  3847  diftpsn3  3849  xpsndisj  5049  funprg  5149  funprgOLD  5150