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Theorem disjsn2 3424
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 3391 . . . 4 (B {A} → B = A)
21eqcomd 2042 . . 3 (B {A} → A = B)
32necon3ai 2248 . 2 (AB → ¬ B {A})
4 disjsn 3423 . 2 (({A} ∩ {B}) = ∅ ↔ ¬ B {A})
53, 4sylibr 137 1 (AB → ({A} ∩ {B}) = ∅)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1242   wcel 1390  wne 2201  cin 2910  c0 3218  {csn 3367
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-fal 1248  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ne 2203  df-ral 2305  df-v 2553  df-dif 2914  df-in 2918  df-nul 3219  df-sn 3373
This theorem is referenced by:  disjpr2  3425  difprsn1  3494  diftpsn3  3496  xpsndisj  4692  funprg  4892  funtp  4895  f1oprg  5111
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