Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > eqneltri | Structured version Visualization version GIF version |
Description: If a class is not an element of another class, an equal class is also not an element. (Contributed by Glauco Siliprandi, 3-Jan-2021.) |
Ref | Expression |
---|---|
eqneltri.1 | ⊢ 𝐴 = 𝐵 |
eqneltri.2 | ⊢ ¬ 𝐵 ∈ 𝐶 |
Ref | Expression |
---|---|
eqneltri | ⊢ ¬ 𝐴 ∈ 𝐶 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqneltri.2 | . 2 ⊢ ¬ 𝐵 ∈ 𝐶 | |
2 | eqneltri.1 | . . 3 ⊢ 𝐴 = 𝐵 | |
3 | eleq1 2676 | . . 3 ⊢ (𝐴 = 𝐵 → (𝐴 ∈ 𝐶 ↔ 𝐵 ∈ 𝐶)) | |
4 | 2, 3 | ax-mp 5 | . 2 ⊢ (𝐴 ∈ 𝐶 ↔ 𝐵 ∈ 𝐶) |
5 | 1, 4 | mtbir 312 | 1 ⊢ ¬ 𝐴 ∈ 𝐶 |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 195 = wceq 1475 ∈ wcel 1977 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-an 385 df-ex 1696 df-cleq 2603 df-clel 2606 |
This theorem is referenced by: eliuniincex 38323 eliincex 38324 salgencntex 39237 |
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