Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > eqneltrrd | GIF version |
Description: If a class is not an element of another class, an equal class is also not an element. Deduction form. (Contributed by David Moews, 1-May-2017.) |
Ref | Expression |
---|---|
eqneltrrd.1 | ⊢ (𝜑 → 𝐴 = 𝐵) |
eqneltrrd.2 | ⊢ (𝜑 → ¬ 𝐴 ∈ 𝐶) |
Ref | Expression |
---|---|
eqneltrrd | ⊢ (𝜑 → ¬ 𝐵 ∈ 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqneltrrd.2 | . 2 ⊢ (𝜑 → ¬ 𝐴 ∈ 𝐶) | |
2 | eqneltrrd.1 | . . 3 ⊢ (𝜑 → 𝐴 = 𝐵) | |
3 | 2 | eleq1d 2106 | . 2 ⊢ (𝜑 → (𝐴 ∈ 𝐶 ↔ 𝐵 ∈ 𝐶)) |
4 | 1, 3 | mtbid 597 | 1 ⊢ (𝜑 → ¬ 𝐵 ∈ 𝐶) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1243 ∈ wcel 1393 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-5 1336 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-4 1400 ax-17 1419 ax-ial 1427 ax-ext 2022 |
This theorem depends on definitions: df-bi 110 df-cleq 2033 df-clel 2036 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |