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

Theorem elnelall 3044
Description: A contradiction concerning membership implies anything. (Contributed by Alexander van der Vekens, 25-Jan-2018.)
Assertion
Ref Expression
elnelall (𝐴𝐵 → (𝐴𝐵𝜑))

Proof of Theorem elnelall
StepHypRef Expression
1 df-nel 3032 . 2 (𝐴𝐵 ↔ ¬ 𝐴𝐵)
2 pm2.24 121 . 2 (𝐴𝐵 → (¬ 𝐴𝐵𝜑))
31, 2syl5bi 232 1 (𝐴𝐵 → (𝐴𝐵𝜑))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wcel 2135  wnel 3031
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8
This theorem depends on definitions:  df-bi 197  df-nel 3032
This theorem is referenced by:  xnn0lenn0nn0  12264
  Copyright terms: Public domain W3C validator