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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > nel2nelini | Structured version Visualization version GIF version |
Description: Membership in an intersection implies membership in the second set. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
Ref | Expression |
---|---|
nel2nelini.1 | ⊢ ¬ 𝐴 ∈ 𝐶 |
Ref | Expression |
---|---|
nel2nelini | ⊢ ¬ 𝐴 ∈ (𝐵 ∩ 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nel2nelini.1 | . 2 ⊢ ¬ 𝐴 ∈ 𝐶 | |
2 | nel2nelin 42140 | . 2 ⊢ (¬ 𝐴 ∈ 𝐶 → ¬ 𝐴 ∈ (𝐵 ∩ 𝐶)) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ ¬ 𝐴 ∈ (𝐵 ∩ 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∈ wcel 2112 ∩ cin 3853 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1912 ax-6 1971 ax-7 2016 ax-8 2114 ax-9 2122 ax-ext 2730 |
This theorem depends on definitions: df-bi 210 df-an 401 df-tru 1542 df-ex 1783 df-sb 2071 df-clab 2737 df-cleq 2751 df-clel 2831 df-v 3409 df-in 3861 |
This theorem is referenced by: (None) |
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