Metamath Proof Explorer
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)
|
|
Ref |
Expression |
|
Hypotheses |
elnelneq2d.1 |
⊢ ( 𝜑 → 𝐴 ∈ 𝐶 ) |
|
|
elnelneq2d.2 |
⊢ ( 𝜑 → ¬ 𝐵 ∈ 𝐶 ) |
|
Assertion |
elnelneq2d |
⊢ ( 𝜑 → ¬ 𝐴 = 𝐵 ) |
Proof
Step |
Hyp |
Ref |
Expression |
1 |
|
elnelneq2d.1 |
⊢ ( 𝜑 → 𝐴 ∈ 𝐶 ) |
2 |
|
elnelneq2d.2 |
⊢ ( 𝜑 → ¬ 𝐵 ∈ 𝐶 ) |
3 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝐴 = 𝐵 ) → 𝐴 = 𝐵 ) |
4 |
1
|
adantr |
⊢ ( ( 𝜑 ∧ 𝐴 = 𝐵 ) → 𝐴 ∈ 𝐶 ) |
5 |
3 4
|
eqeltrrd |
⊢ ( ( 𝜑 ∧ 𝐴 = 𝐵 ) → 𝐵 ∈ 𝐶 ) |
6 |
2 5
|
mtand |
⊢ ( 𝜑 → ¬ 𝐴 = 𝐵 ) |