Theorem elnelneqd 3059

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 486	. . 3 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → 𝐶 ∈ 𝐴)
4		simpr 490	. . 3 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → 𝐴 = 𝐵)
5	3, 4	eleqtrd 2867	. 2 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → 𝐶 ∈ 𝐵)
6	1, 5	mtand 828	1 ⊢ (𝜑 → ¬ 𝐴 = 𝐵)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2146

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 2148  ax-9 2156  ax-ext 2737

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813  df-cleq 2757  df-clel 2840

This theorem is used by:  mnuprdlem1  45059  mnuprdlem2  45060