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Mirrors > Home > ILE Home > Th. List > ssneldd | GIF version |
Description: If an element is not in a 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) |
ssneldd.2 | ⊢ (𝜑 → ¬ 𝐶 ∈ 𝐵) |
Ref | Expression |
---|---|
ssneldd | ⊢ (𝜑 → ¬ 𝐶 ∈ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssneldd.2 | . 2 ⊢ (𝜑 → ¬ 𝐶 ∈ 𝐵) | |
2 | ssneld.1 | . . 3 ⊢ (𝜑 → 𝐴 ⊆ 𝐵) | |
3 | 2 | ssneld 3149 | . 2 ⊢ (𝜑 → (¬ 𝐶 ∈ 𝐵 → ¬ 𝐶 ∈ 𝐴)) |
4 | 1, 3 | mpd 13 | 1 ⊢ (𝜑 → ¬ 𝐶 ∈ 𝐴) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∈ wcel 2141 ⊆ wss 3121 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 609 ax-in2 610 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-11 1499 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-ext 2152 |
This theorem depends on definitions: df-bi 116 df-nf 1454 df-sb 1756 df-clab 2157 df-cleq 2163 df-clel 2166 df-in 3127 df-ss 3134 |
This theorem is referenced by: 0nelrel 4657 addnqprlemfl 7521 addnqprlemfu 7522 mulnqprlemfl 7537 mulnqprlemfu 7538 cauappcvgprlemladdru 7618 fprodntrivap 11547 fprodssdc 11553 |
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