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Theorem elnelne1 3055
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 3045 . 2 (𝐴𝐶 ↔ ¬ 𝐴𝐶)
2 nelne1 3037 . 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 2106  wne 2938  wnel 3044
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1777  df-cleq 2727  df-clel 2814  df-ne 2939  df-nel 3045
This theorem is referenced by:  sticksstones1  42128
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