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Mirrors > Home > MPE Home > Th. List > Mathboxes > ssinss2d | Structured version Visualization version GIF version |
Description: Intersection preserves subclass relationship. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
Ref | Expression |
---|---|
ssinss2d.1 | ⊢ (𝜑 → 𝐵 ⊆ 𝐶) |
Ref | Expression |
---|---|
ssinss2d | ⊢ (𝜑 → (𝐴 ∩ 𝐵) ⊆ 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | incom 4177 | . 2 ⊢ (𝐴 ∩ 𝐵) = (𝐵 ∩ 𝐴) | |
2 | ssinss2d.1 | . . 3 ⊢ (𝜑 → 𝐵 ⊆ 𝐶) | |
3 | 2 | ssinss1d 41303 | . 2 ⊢ (𝜑 → (𝐵 ∩ 𝐴) ⊆ 𝐶) |
4 | 1, 3 | eqsstrid 4014 | 1 ⊢ (𝜑 → (𝐴 ∩ 𝐵) ⊆ 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∩ cin 3934 ⊆ wss 3935 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-rab 3147 df-v 3496 df-in 3942 df-ss 3951 |
This theorem is referenced by: caragenuncllem 42788 |
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