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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > ssinss1d | Structured version Visualization version GIF version |
Description: Intersection preserves subclass relationship. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
Ref | Expression |
---|---|
ssinss1d.1 | ⊢ (𝜑 → 𝐴 ⊆ 𝐶) |
Ref | Expression |
---|---|
ssinss1d | ⊢ (𝜑 → (𝐴 ∩ 𝐵) ⊆ 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssinss1d.1 | . 2 ⊢ (𝜑 → 𝐴 ⊆ 𝐶) | |
2 | ssinss1 4036 | . 2 ⊢ (𝐴 ⊆ 𝐶 → (𝐴 ∩ 𝐵) ⊆ 𝐶) | |
3 | 1, 2 | syl 17 | 1 ⊢ (𝜑 → (𝐴 ∩ 𝐵) ⊆ 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∩ cin 3767 ⊆ wss 3768 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-ext 2776 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-clab 2785 df-cleq 2791 df-clel 2794 df-nfc 2929 df-v 3386 df-in 3775 df-ss 3782 |
This theorem is referenced by: ssinss2d 39976 ovolsplit 40937 caragenuncllem 41461 carageniuncllem1 41470 ovnsplit 41597 vonvolmbllem 41609 vonvolmbl 41610 |
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