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Mirrors > Home > MPE Home > Th. List > 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 4245 | . 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 3949 ⊆ wss 3950 |
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 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2707 |
This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1543 df-ex 1780 df-sb 2065 df-clab 2714 df-cleq 2728 df-clel 2815 df-v 3481 df-in 3957 df-ss 3967 |
This theorem is referenced by: exsslsb 33634 ssinss2d 45038 ovolsplit 45976 caragenuncllem 46500 carageniuncllem1 46509 ovnsplit 46636 vonvolmbllem 46648 vonvolmbl 46649 |
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