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Theorem in0 3252
 Description: The intersection of a class with the empty set is the empty set. Theorem 16 of [Suppes] p. 26. (Contributed by NM, 5-Aug-1993.)
Assertion
Ref Expression
in0 (𝐴 ∩ ∅) = ∅

Proof of Theorem in0
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 noel 3228 . . . 4 ¬ 𝑥 ∈ ∅
21bianfi 854 . . 3 (𝑥 ∈ ∅ ↔ (𝑥𝐴𝑥 ∈ ∅))
32bicomi 123 . 2 ((𝑥𝐴𝑥 ∈ ∅) ↔ 𝑥 ∈ ∅)
43ineqri 3130 1 (𝐴 ∩ ∅) = ∅
 Colors of variables: wff set class Syntax hints:   ∧ wa 97   = wceq 1243   ∈ wcel 1393   ∩ cin 2916  ∅c0 3224 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 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022 This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-dif 2920  df-in 2924  df-nul 3225 This theorem is referenced by:  res0  4616
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