Theorem nelaneq 9552

Description: A class is not an element of and equal to a class at the same time. Variant of elneq 9551 analogously to elnotel 9563 and en2lp 9559. (Proposed by BJ, 18-Jun-2022.) (Contributed by AV, 18-Jun-2022.)

Assertion

Ref	Expression
nelaneq	⊢ ¬ (𝐴 ∈ 𝐵 ∧ 𝐴 = 𝐵)

Proof of Theorem nelaneq

Step	Hyp	Ref	Expression
1		elneq 9551	. . 3 ⊢ (𝐴 ∈ 𝐵 → 𝐴 ≠ 𝐵)
2		orc 867	. . . 4 ⊢ (¬ 𝐴 ∈ 𝐵 → (¬ 𝐴 ∈ 𝐵 ∨ ¬ 𝐴 = 𝐵))
3		neneq 2931	. . . . 5 ⊢ (𝐴 ≠ 𝐵 → ¬ 𝐴 = 𝐵)
4	3	olcd 874	. . . 4 ⊢ (𝐴 ≠ 𝐵 → (¬ 𝐴 ∈ 𝐵 ∨ ¬ 𝐴 = 𝐵))
5	2, 4	ja 186	. . 3 ⊢ ((𝐴 ∈ 𝐵 → 𝐴 ≠ 𝐵) → (¬ 𝐴 ∈ 𝐵 ∨ ¬ 𝐴 = 𝐵))
6	1, 5	ax-mp 5	. 2 ⊢ (¬ 𝐴 ∈ 𝐵 ∨ ¬ 𝐴 = 𝐵)
7		ianor 983	. 2 ⊢ (¬ (𝐴 ∈ 𝐵 ∧ 𝐴 = 𝐵) ↔ (¬ 𝐴 ∈ 𝐵 ∨ ¬ 𝐴 = 𝐵))
8	6, 7	mpbir 231	1 ⊢ ¬ (𝐴 ∈ 𝐵 ∧ 𝐴 = 𝐵)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   ∨ wo 847   = wceq 1540   ∈ wcel 2109   ≠ wne 2925

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-12 2178  ax-ext 2701  ax-sep 5251  ax-pr 5387  ax-reg 9545

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1543  df-ex 1780  df-nf 1784  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-ne 2926  df-ral 3045  df-rex 3054  df-v 3449  df-un 3919  df-sn 4590  df-pr 4592

This theorem is referenced by:  epinid0  9553