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Theorem elnelall 40210
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 2687 . 2 (𝐴𝐵 ↔ ¬ 𝐴𝐵)
2 pm2.24 119 . 2 (𝐴𝐵 → (¬ 𝐴𝐵𝜑))
31, 2syl5bi 230 1 (𝐴𝐵 → (𝐴𝐵𝜑))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wcel 1938  wnel 2685
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 195  df-nel 2687
This theorem is referenced by: (None)
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