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Theorem int0el 4911
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 4895 . 2 (∅ ∈ 𝐴 𝐴 ⊆ ∅)
2 0ss 4331 . . 3 ∅ ⊆ 𝐴
32a1i 11 . 2 (∅ ∈ 𝐴 → ∅ ⊆ 𝐴)
41, 3eqssd 3938 1 (∅ ∈ 𝐴 𝐴 = ∅)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1539  wcel 2106  wss 3887  c0 4257   cint 4880
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709
This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-v 3432  df-dif 3890  df-in 3894  df-ss 3904  df-nul 4258  df-int 4881
This theorem is referenced by:  intv  5285  inton  6317  onint0  7632  oev2  8341  ipolub00  46235
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