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Theorem nel02 34646
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 4149 . 2 ¬ 𝐵 ∈ ∅
2 eleq2 2896 . 2 (𝐴 = ∅ → (𝐵𝐴𝐵 ∈ ∅))
31, 2mtbiri 319 1 (𝐴 = ∅ → ¬ 𝐵𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1658  wcel 2166  c0 4145
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-ext 2804
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-clab 2813  df-cleq 2819  df-clel 2822  df-nfc 2959  df-v 3417  df-dif 3802  df-nul 4146
This theorem is referenced by: (None)
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