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Mathbox for Rohan Ridenour |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > elnelneq2d | Structured version Visualization version GIF version |
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 |
---|---|
elnelneq2d.1 | ⊢ (𝜑 → 𝐴 ∈ 𝐶) |
elnelneq2d.2 | ⊢ (𝜑 → ¬ 𝐵 ∈ 𝐶) |
Ref | Expression |
---|---|
elnelneq2d | ⊢ (𝜑 → ¬ 𝐴 = 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elnelneq2d.2 | . 2 ⊢ (𝜑 → ¬ 𝐵 ∈ 𝐶) | |
2 | simpr 484 | . . 3 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → 𝐴 = 𝐵) | |
3 | elnelneq2d.1 | . . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝐶) | |
4 | 3 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → 𝐴 ∈ 𝐶) |
5 | 2, 4 | eqeltrrd 2833 | . 2 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → 𝐵 ∈ 𝐶) |
6 | 1, 5 | mtand 813 | 1 ⊢ (𝜑 → ¬ 𝐴 = 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2105 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-ext 2702 |
This theorem depends on definitions: df-bi 206 df-an 396 df-ex 1781 df-cleq 2723 df-clel 2809 |
This theorem is referenced by: mnurndlem1 43505 |
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