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Theorem nelaneq 8746
Description: A class is not an element of and equal to a class at the same time. Variant of elneq 8745 analogously to elnotel 8755 and en2lp 8752. (Proposed by BJ, 18-Jun-2022.) (Contributed by AV, 18-Jun-2022.)
Assertion
Ref Expression
nelaneq ¬ (𝐴𝐵𝐴 = 𝐵)

Proof of Theorem nelaneq
StepHypRef Expression
1 elneq 8745 . . 3 (𝐴𝐵𝐴𝐵)
2 orc 894 . . . 4 𝐴𝐵 → (¬ 𝐴𝐵 ∨ ¬ 𝐴 = 𝐵))
3 neneq 2977 . . . . 5 (𝐴𝐵 → ¬ 𝐴 = 𝐵)
43olcd 901 . . . 4 (𝐴𝐵 → (¬ 𝐴𝐵 ∨ ¬ 𝐴 = 𝐵))
52, 4ja 175 . . 3 ((𝐴𝐵𝐴𝐵) → (¬ 𝐴𝐵 ∨ ¬ 𝐴 = 𝐵))
61, 5ax-mp 5 . 2 𝐴𝐵 ∨ ¬ 𝐴 = 𝐵)
7 ianor 1005 . 2 (¬ (𝐴𝐵𝐴 = 𝐵) ↔ (¬ 𝐴𝐵 ∨ ¬ 𝐴 = 𝐵))
86, 7mpbir 223 1 ¬ (𝐴𝐵𝐴 = 𝐵)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 385  wo 874   = wceq 1653  wcel 2157  wne 2971
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-sep 4975  ax-nul 4983  ax-pr 5097  ax-reg 8739
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ne 2972  df-ral 3094  df-rex 3095  df-v 3387  df-dif 3772  df-un 3774  df-nul 4116  df-sn 4369  df-pr 4371
This theorem is referenced by:  epinid0  8747
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