Mathbox for Glauco Siliprandi |
< Previous
Next >
Nearby theorems |
||
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 3927 | . 2 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐶 ∈ 𝐴) → 𝐶 ∈ 𝐵) | |
2 | 1 | stoic1a 1773 | 1 ⊢ ((𝐴 ⊆ 𝐵 ∧ ¬ 𝐶 ∈ 𝐵) → ¬ 𝐶 ∈ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 396 ∈ wcel 2105 ⊆ wss 3898 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-ext 2707 |
This theorem depends on definitions: df-bi 206 df-an 397 df-tru 1543 df-ex 1781 df-sb 2067 df-clab 2714 df-cleq 2728 df-clel 2814 df-v 3443 df-in 3905 df-ss 3915 |
This theorem is referenced by: nelrnres 43061 supminfxr2 43353 |
Copyright terms: Public domain | W3C validator |