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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 3069 | . 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 1465 ⊆ wss 3041 |
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 588 ax-in2 589 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-11 1469 ax-4 1472 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 |
This theorem depends on definitions: df-bi 116 df-nf 1422 df-sb 1721 df-clab 2104 df-cleq 2110 df-clel 2113 df-in 3047 df-ss 3054 |
This theorem is referenced by: 0nelrel 4555 addnqprlemfl 7335 addnqprlemfu 7336 mulnqprlemfl 7351 mulnqprlemfu 7352 cauappcvgprlemladdru 7432 |
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