Theorem nelaneqOLDOLD 9591

Description: Obsolete version of nelaneq 9589 as of 31-Dec-2025. (Proposed by BJ, 18-Jun-2022.) (Contributed by AV, 18-Jun-2022.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

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

Proof of Theorem nelaneqOLDOLD

Step	Hyp	Ref	Expression
1		elneq 9588	. . 3 ⊢ (𝐴 ∈ 𝐵 → 𝐴 ≠ 𝐵)
2		orc 881	. . . 4 ⊢ (¬ 𝐴 ∈ 𝐵 → (¬ 𝐴 ∈ 𝐵 ∨ ¬ 𝐴 = 𝐵))
3		neneq 2962	. . . . 5 ⊢ (𝐴 ≠ 𝐵 → ¬ 𝐴 = 𝐵)
4	3	olcd 888	. . . 4 ⊢ (𝐴 ≠ 𝐵 → (¬ 𝐴 ∈ 𝐵 ∨ ¬ 𝐴 = 𝐵))
5	2, 4	ja 188	. . 3 ⊢ ((𝐴 ∈ 𝐵 → 𝐴 ≠ 𝐵) → (¬ 𝐴 ∈ 𝐵 ∨ ¬ 𝐴 = 𝐵))
6	1, 5	ax-mp 5	. 2 ⊢ (¬ 𝐴 ∈ 𝐵 ∨ ¬ 𝐴 = 𝐵)
7		ianor 997	. 2 ⊢ (¬ (𝐴 ∈ 𝐵 ∧ 𝐴 = 𝐵) ↔ (¬ 𝐴 ∈ 𝐵 ∨ ¬ 𝐴 = 𝐵))
8	6, 7	mpbir 234	1 ⊢ ¬ (𝐴 ∈ 𝐵 ∧ 𝐴 = 𝐵)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   ∨ wo 861   = wceq 1570   ∈ wcel 2145   ≠ wne 2956

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2733  ax-sep 5249  ax-reg 9579

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2740  df-cleq 2753  df-clel 2836  df-ne 2957

This theorem is used by: (None)