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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 4236 | . 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 3943 ⊆ wss 3944 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2696 |
This theorem depends on definitions: df-bi 206 df-an 395 df-tru 1536 df-ex 1774 df-sb 2060 df-clab 2703 df-cleq 2717 df-clel 2802 df-v 3463 df-in 3951 df-ss 3961 |
This theorem is referenced by: ssinss2d 44563 ovolsplit 45511 caragenuncllem 46035 carageniuncllem1 46044 ovnsplit 46171 vonvolmbllem 46183 vonvolmbl 46184 |
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