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Mirrors > Home > MPE Home > Th. List > inss | Structured version Visualization version GIF version |
Description: Inclusion of an intersection of two classes. (Contributed by NM, 30-Oct-2014.) |
Ref | Expression |
---|---|
inss | ⊢ ((𝐴 ⊆ 𝐶 ∨ 𝐵 ⊆ 𝐶) → (𝐴 ∩ 𝐵) ⊆ 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssinss1 4164 | . 2 ⊢ (𝐴 ⊆ 𝐶 → (𝐴 ∩ 𝐵) ⊆ 𝐶) | |
2 | incom 4128 | . . 3 ⊢ (𝐴 ∩ 𝐵) = (𝐵 ∩ 𝐴) | |
3 | ssinss1 4164 | . . 3 ⊢ (𝐵 ⊆ 𝐶 → (𝐵 ∩ 𝐴) ⊆ 𝐶) | |
4 | 2, 3 | eqsstrid 3963 | . 2 ⊢ (𝐵 ⊆ 𝐶 → (𝐴 ∩ 𝐵) ⊆ 𝐶) |
5 | 1, 4 | jaoi 854 | 1 ⊢ ((𝐴 ⊆ 𝐶 ∨ 𝐵 ⊆ 𝐶) → (𝐴 ∩ 𝐵) ⊆ 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∨ wo 844 ∩ cin 3880 ⊆ wss 3881 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-ext 2770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-tru 1541 df-ex 1782 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-rab 3115 df-v 3443 df-in 3888 df-ss 3898 |
This theorem is referenced by: pmatcoe1fsupp 21306 ppttop 21612 inindif 30287 disjorimxrn 36138 iunrelexp0 40403 ntrclsk3 40773 icccncfext 42529 |
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