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Theorem elnelne2 2937
Description: Two classes are different if they don't belong to the same class. (Contributed by AV, 28-Jan-2020.)
Assertion
Ref Expression
elnelne2 ((𝐴𝐶𝐵𝐶) → 𝐴𝐵)

Proof of Theorem elnelne2
StepHypRef Expression
1 df-nel 2927 . 2 (𝐵𝐶 ↔ ¬ 𝐵𝐶)
2 nelne2 2920 . 2 ((𝐴𝐶 ∧ ¬ 𝐵𝐶) → 𝐴𝐵)
31, 2sylan2b 491 1 ((𝐴𝐶𝐵𝐶) → 𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 383  wcel 2030  wne 2823  wnel 2926
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-ext 2631
This theorem depends on definitions:  df-bi 197  df-an 385  df-ex 1745  df-cleq 2644  df-clel 2647  df-ne 2824  df-nel 2927
This theorem is referenced by:  nelrnfvne  6393  eldmrexrnb  6406  absprodnn  15378  frgrncvvdeqlem2  27280  frgrncvvdeqlem3  27281  afv0nbfvbi  41552  2zrngnmlid  42274  2zrngnmrid  42275
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