Theorem elnelneqd 44739

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 484	. . 3 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → 𝐶 ∈ 𝐴)
4		simpr 488	. . 3 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → 𝐴 = 𝐵)
5	3, 4	eleqtrd 2863	. 2 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → 𝐶 ∈ 𝐵)
6	1, 5	mtand 825	1 ⊢ (𝜑 → ¬ 𝐴 = 𝐵)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 399   = wceq 1559   ∈ wcel 2141

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-ext 2733

This theorem depends on definitions:  df-bi 209  df-an 400  df-ex 1799  df-cleq 2753  df-clel 2836

This theorem is referenced by:  mnuprdlem1  44809  mnuprdlem2  44810