Theorem elnelneqd 43036

Description: Two classes are not equal if there is an element of one which is not an element of the other. (Contributed by Rohan Ridenour, 11-Aug-2023.)

Hypotheses

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

Assertion

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

Proof of Theorem elnelneqd

Step	Hyp	Ref	Expression
1		elnelneqd.2	. 2 ⊢ (𝜑 → ¬ 𝐶 ∈ 𝐵)
2		elnelneqd.1	. . . 4 ⊢ (𝜑 → 𝐶 ∈ 𝐴)
3	2	adantr 481	. . 3 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → 𝐶 ∈ 𝐴)
4		simpr 485	. . 3 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → 𝐴 = 𝐵)
5	3, 4	eleqtrd 2835	. 2 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → 𝐶 ∈ 𝐵)
6	1, 5	mtand 814	1 ⊢ (𝜑 → ¬ 𝐴 = 𝐵)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 396   = wceq 1541   ∈ wcel 2106

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703

This theorem depends on definitions:  df-bi 206  df-an 397  df-ex 1782  df-cleq 2724  df-clel 2810

This theorem is referenced by:  mnuprdlem1  43113  mnuprdlem2  43114