Theorem elnelneq2d 3059

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 490	. . 3 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → 𝐴 = 𝐵)
3		elnelneq2d.1	. . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝐶)
4	3	adantr 486	. . 3 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → 𝐴 ∈ 𝐶)
5	2, 4	eqeltrrd 2865	. 2 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → 𝐵 ∈ 𝐶)
6	1, 5	mtand 828	1 ⊢ (𝜑 → ¬ 𝐴 = 𝐵)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2147

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813  df-cleq 2755  df-clel 2839

This theorem is used by:  tgaaddcpbllem2  29377  prlngmolem1  29457  mnurndlem1  45455