Theorem elnelneq2d 43531

Description: Two classes are not equal if one but not the other is an element of a given class. (Contributed by Rohan Ridenour, 12-Aug-2023.)

Hypotheses

Ref	Expression
elnelneq2d.1	⊢ (𝜑 → 𝐴 ∈ 𝐶)
elnelneq2d.2	⊢ (𝜑 → ¬ 𝐵 ∈ 𝐶)

Assertion

Ref	Expression
elnelneq2d	⊢ (𝜑 → ¬ 𝐴 = 𝐵)

Proof of Theorem elnelneq2d

Step	Hyp	Ref	Expression
1		elnelneq2d.2	. 2 ⊢ (𝜑 → ¬ 𝐵 ∈ 𝐶)
2		simpr 484	. . 3 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → 𝐴 = 𝐵)
3		elnelneq2d.1	. . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝐶)
4	3	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → 𝐴 ∈ 𝐶)
5	2, 4	eqeltrrd 2828	. 2 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → 𝐵 ∈ 𝐶)
6	1, 5	mtand 813	1 ⊢ (𝜑 → ¬ 𝐴 = 𝐵)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1533   ∈ wcel 2098

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 2697

This theorem depends on definitions:  df-bi 206  df-an 396  df-ex 1774  df-cleq 2718  df-clel 2804

This theorem is referenced by:  mnurndlem1  43616