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Theorem elnelne1 3046
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 3037 . 2 (𝐴𝐶 ↔ ¬ 𝐴𝐶)
2 nelne1 3029 . 2 ((𝐴𝐵 ∧ ¬ 𝐴𝐶) → 𝐵𝐶)
31, 2sylan2b 493 1 ((𝐴𝐵𝐴𝐶) → 𝐵𝐶)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 383  wcel 2140  wne 2933  wnel 3036
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1989  ax-6 2055  ax-7 2091  ax-9 2149  ax-ext 2741
This theorem depends on definitions:  df-bi 197  df-an 385  df-ex 1854  df-cleq 2754  df-clel 2757  df-ne 2934  df-nel 3037
This theorem is referenced by: (None)
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