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Theorem nel02 34440
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 4067 . 2 ¬ 𝐵 ∈ ∅
2 eleq2 2839 . 2 (𝐴 = ∅ → (𝐵𝐴𝐵 ∈ ∅))
31, 2mtbiri 316 1 (𝐴 = ∅ → ¬ 𝐵𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1631  wcel 2145  c0 4063
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-v 3353  df-dif 3726  df-nul 4064
This theorem is referenced by: (None)
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