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Theorem 0in 3403
 Description: The intersection of the empty set with a class is the empty set. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
Assertion
Ref Expression
0in (∅ ∩ 𝐴) = ∅

Proof of Theorem 0in
StepHypRef Expression
1 incom 3273 . 2 (∅ ∩ 𝐴) = (𝐴 ∩ ∅)
2 in0 3402 . 2 (𝐴 ∩ ∅) = ∅
31, 2eqtri 2161 1 (∅ ∩ 𝐴) = ∅
 Colors of variables: wff set class Syntax hints:   = wceq 1332   ∩ cin 3075  ∅c0 3368 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122 This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-v 2691  df-dif 3078  df-in 3082  df-nul 3369 This theorem is referenced by:  setsfun  12034  setsfun0  12035  restsn  12389
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