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Theorem in0 3397
 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 3367 . . . 4 ¬ 𝑥 ∈ ∅
21bianfi 931 . . 3 (𝑥 ∈ ∅ ↔ (𝑥𝐴𝑥 ∈ ∅))
32bicomi 131 . 2 ((𝑥𝐴𝑥 ∈ ∅) ↔ 𝑥 ∈ ∅)
43ineqri 3269 1 (𝐴 ∩ ∅) = ∅
 Colors of variables: wff set class Syntax hints:   ∧ wa 103   = wceq 1331   ∈ wcel 1480   ∩ cin 3070  ∅c0 3363 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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121 This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-v 2688  df-dif 3073  df-in 3077  df-nul 3364 This theorem is referenced by:  0in  3398  res0  4823  dju0en  7075  rest0  12362
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