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Theorem int0el 4473
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 4457 . 2 (∅ ∈ 𝐴 𝐴 ⊆ ∅)
2 0ss 3944 . . 3 ∅ ⊆ 𝐴
32a1i 11 . 2 (∅ ∈ 𝐴 → ∅ ⊆ 𝐴)
41, 3eqssd 3600 1 (∅ ∈ 𝐴 𝐴 = ∅)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1480  wcel 1987  wss 3555  c0 3891   cint 4440
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-v 3188  df-dif 3558  df-in 3562  df-ss 3569  df-nul 3892  df-int 4441
This theorem is referenced by:  intv  4801  inton  5741  onint0  6943  oev2  7548
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