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Theorem disjsn2 3787
 Description: Intersection of distinct singletons is disjoint. (Contributed by NM, 25-May-1998.)
Assertion
Ref Expression
disjsn2 (AB → ({A} ∩ {B}) = )

Proof of Theorem disjsn2
StepHypRef Expression
1 elsni 3757 . . . 4 (B {A} → B = A)
21eqcomd 2358 . . 3 (B {A} → A = B)
32necon3ai 2556 . 2 (AB → ¬ B {A})
4 disjsn 3786 . 2 (({A} ∩ {B}) = ↔ ¬ B {A})
53, 4sylibr 203 1 (AB → ({A} ∩ {B}) = )
 Colors of variables: wff setvar class 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-1 5  ax-2 6  ax-3 7  ax-mp 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
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