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Theorem elnelne1 3057
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 3047 . 2 (𝐴𝐶 ↔ ¬ 𝐴𝐶)
2 nelne1 3039 . 2 ((𝐴𝐵 ∧ ¬ 𝐴𝐶) → 𝐵𝐶)
31, 2sylan2b 594 1 ((𝐴𝐵𝐴𝐶) → 𝐵𝐶)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395  wcel 2108  wne 2940  wnel 3046
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1780  df-cleq 2729  df-clel 2816  df-ne 2941  df-nel 3047
This theorem is referenced by:  sticksstones1  42147
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