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Theorem elnelne1 2892
Description: Two classes are different if they don't contain the same element. (Contributed by AV, 28-Jan-2020.)
Assertion
Ref Expression
elnelne1 ((𝐴𝐵𝐴𝐶) → 𝐵𝐶)

Proof of Theorem elnelne1
StepHypRef Expression
1 df-nel 2782 . 2 (𝐴𝐶 ↔ ¬ 𝐴𝐶)
2 nelne1 2877 . 2 ((𝐴𝐵 ∧ ¬ 𝐴𝐶) → 𝐵𝐶)
31, 2sylan2b 490 1 ((𝐴𝐵𝐴𝐶) → 𝐵𝐶)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 382  wcel 1976  wne 2779  wnel 2780
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-ext 2589
This theorem depends on definitions:  df-bi 195  df-an 384  df-ex 1695  df-cleq 2602  df-clel 2605  df-ne 2781  df-nel 2782
This theorem is referenced by: (None)
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