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 41421 | . 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 2113 ∩ cin 3938 |
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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-v 3499 df-in 3946 |
This theorem is referenced by: (None) |
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