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Mirrors > Home > ILE Home > Th. List > ssneld | GIF version |
Description: If a class is not in another class, it is also not in a subclass of that class. Deduction form. (Contributed by David Moews, 1-May-2017.) |
Ref | Expression |
---|---|
ssneld.1 | ⊢ (𝜑 → 𝐴 ⊆ 𝐵) |
Ref | Expression |
---|---|
ssneld | ⊢ (𝜑 → (¬ 𝐶 ∈ 𝐵 → ¬ 𝐶 ∈ 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssneld.1 | . . 3 ⊢ (𝜑 → 𝐴 ⊆ 𝐵) | |
2 | 1 | sseld 3011 | . 2 ⊢ (𝜑 → (𝐶 ∈ 𝐴 → 𝐶 ∈ 𝐵)) |
3 | 2 | con3d 594 | 1 ⊢ (𝜑 → (¬ 𝐶 ∈ 𝐵 → ¬ 𝐶 ∈ 𝐴)) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∈ wcel 1436 ⊆ wss 2986 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 577 ax-in2 578 ax-5 1379 ax-7 1380 ax-gen 1381 ax-ie1 1425 ax-ie2 1426 ax-8 1438 ax-11 1440 ax-4 1443 ax-17 1462 ax-i9 1466 ax-ial 1470 ax-i5r 1471 ax-ext 2067 |
This theorem depends on definitions: df-bi 115 df-nf 1393 df-sb 1690 df-clab 2072 df-cleq 2078 df-clel 2081 df-in 2992 df-ss 2999 |
This theorem is referenced by: ssneldd 3015 sumdc 10583 decidin 11054 |
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