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Theorem int0el 3645
 Description: The intersection of a class containing the empty set is empty. (Contributed by NM, 24-Apr-2004.)
Assertion
Ref Expression
int0el (∅ ∈ 𝐴 𝐴 = ∅)

Proof of Theorem int0el
StepHypRef Expression
1 intss1 3630 . 2 (∅ ∈ 𝐴 𝐴 ⊆ ∅)
2 0ss 3255 . . 3 ∅ ⊆ 𝐴
32a1i 9 . 2 (∅ ∈ 𝐴 → ∅ ⊆ 𝐴)
41, 3eqssd 2962 1 (∅ ∈ 𝐴 𝐴 = ∅)
 Colors of variables: wff set class Syntax hints:   → wi 4   = wceq 1243   ∈ wcel 1393   ⊆ wss 2917  ∅c0 3224  ∩ cint 3615 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-ss 2931  df-nul 3225  df-int 3616 This theorem is referenced by:  intv  3923  inton  4130
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