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Theorem int0el 4979
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 4963 . 2 (∅ ∈ 𝐴 𝐴 ⊆ ∅)
2 0ss 4400 . . 3 ∅ ⊆ 𝐴
32a1i 11 . 2 (∅ ∈ 𝐴 → ∅ ⊆ 𝐴)
41, 3eqssd 4001 1 (∅ ∈ 𝐴 𝐴 = ∅)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1540  wcel 2108  wss 3951  c0 4333   cint 4946
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-v 3482  df-dif 3954  df-ss 3968  df-nul 4334  df-int 4947
This theorem is referenced by:  intv  5364  inton  6442  onint0  7811  oev2  8561  cuteq0  27877  ipolub00  48882
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