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Theorem elnelne2 3050
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 3039 . 2 (𝐵𝐶 ↔ ¬ 𝐵𝐶)
2 nelne2 3032 . 2 ((𝐴𝐶 ∧ ¬ 𝐵𝐶) → 𝐴𝐵)
31, 2sylan2b 593 1 ((𝐴𝐶𝐵𝐶) → 𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395  wcel 2098  wne 2932  wnel 3038
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2695
This theorem depends on definitions:  df-bi 206  df-an 396  df-ex 1774  df-cleq 2716  df-clel 2802  df-ne 2933  df-nel 3039
This theorem is referenced by:  nelrnfvne  7069  eldmrexrnb  7083  absprodnn  16552  frgrncvvdeqlem2  30022  frgrncvvdeqlem3  30023  afv0nbfvbi  46344  uniimaelsetpreimafv  46549  imasetpreimafvbijlemfv1  46556  2zrngnmlid  47118  2zrngnmrid  47119
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