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Theorem nel02 4270
Description: The empty set has no elements. (Contributed by Peter Mazsa, 4-Jan-2018.)
Assertion
Ref Expression
nel02 (𝐴 = ∅ → ¬ 𝐵𝐴)

Proof of Theorem nel02
StepHypRef Expression
1 noel 4269 . 2 ¬ 𝐵 ∈ ∅
2 eleq2 2902 . 2 (𝐴 = ∅ → (𝐵𝐴𝐵 ∈ ∅))
31, 2mtbiri 330 1 (𝐴 = ∅ → ¬ 𝐵𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1538  wcel 2114  c0 4265
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-ext 2794
This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-sb 2070  df-clab 2801  df-cleq 2815  df-clel 2894  df-dif 3911  df-nul 4266
This theorem is referenced by:  iresn0n0  5901  0mpo0  7221  disjxun0  30332
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