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Theorem nel02 4362
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 4360 . 2 ¬ 𝐵 ∈ ∅
2 eleq2 2833 . 2 (𝐴 = ∅ → (𝐵𝐴𝐵 ∈ ∅))
31, 2mtbiri 327 1 (𝐴 = ∅ → ¬ 𝐵𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1537  wcel 2108  c0 4352
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-dif 3979  df-nul 4353
This theorem is referenced by:  iresn0n0  6083  0mpo0  7533  nbgr0vtx  29390  disjxun0  32596  disjlem14  38754  oe0rif  43247  clnbgr0vtx  47708
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