MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  nel02 Structured version   Visualization version   GIF version

Theorem nel02 4268
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 4266 . 2 ¬ 𝐵 ∈ ∅
2 eleq2 2827 . 2 (𝐴 = ∅ → (𝐵𝐴𝐵 ∈ ∅))
31, 2mtbiri 327 1 (𝐴 = ∅ → ¬ 𝐵𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1539  wcel 2106  c0 4258
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709
This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-dif 3891  df-nul 4259
This theorem is referenced by:  iresn0n0  5965  0mpo0  7358  disjxun0  30910
  Copyright terms: Public domain W3C validator