Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > nel1nelini | Structured version Visualization version GIF version |
Description: Membership in an intersection implies membership in the first set. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
Ref | Expression |
---|---|
nel1nelini.1 | ⊢ ¬ 𝐴 ∈ 𝐵 |
Ref | Expression |
---|---|
nel1nelini | ⊢ ¬ 𝐴 ∈ (𝐵 ∩ 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nel1nelini.1 | . 2 ⊢ ¬ 𝐴 ∈ 𝐵 | |
2 | nel1nelin 42368 | . 2 ⊢ (¬ 𝐴 ∈ 𝐵 → ¬ 𝐴 ∈ (𝐵 ∩ 𝐶)) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ ¬ 𝐴 ∈ (𝐵 ∩ 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∈ wcel 2110 ∩ cin 3865 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-ext 2708 |
This theorem depends on definitions: df-bi 210 df-an 400 df-tru 1546 df-ex 1788 df-sb 2071 df-clab 2715 df-cleq 2729 df-clel 2816 df-v 3410 df-in 3873 |
This theorem is referenced by: (None) |
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