| Mathbox for Glauco Siliprandi |
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| Mirrors > Home > MPE Home > Th. List > Mathboxes > ssnel | Structured version Visualization version GIF version | ||
| Description: If not element of a set, then not element of a subset. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
| Ref | Expression |
|---|---|
| ssnel | ⊢ ((𝐴 ⊆ 𝐵 ∧ ¬ 𝐶 ∈ 𝐵) → ¬ 𝐶 ∈ 𝐴) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | ssel2 3978 | . 2 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐶 ∈ 𝐴) → 𝐶 ∈ 𝐵) | |
| 2 | 1 | stoic1a 1772 | 1 ⊢ ((𝐴 ⊆ 𝐵 ∧ ¬ 𝐶 ∈ 𝐵) → ¬ 𝐶 ∈ 𝐴) |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 ∈ wcel 2108 ⊆ wss 3951 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1780 df-clel 2816 df-ss 3968 |
| This theorem is referenced by: nelrnres 45192 supminfxr2 45480 |
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