Theorem disjsn2 3788

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 3758	. . . 4 ⊢ (B ∈ {A} → B = A)
2	1	eqcomd 2358	. . 3 ⊢ (B ∈ {A} → A = B)
3	2	necon3ai 2557	. 2 ⊢ (A ≠ B → ¬ B ∈ {A})
4		disjsn 3787	. 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 2517   ∩ cin 3209  ∅c0 3551  {csn 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 2479  df-ne 2519  df-ral 2620  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-dif 3216  df-nul 3552  df-sn 3742

This theorem is referenced by:  difprsn1  3848  diftpsn3  3850  xpsndisj  5050  funprg  5150  funprgOLD  5151