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Mirrors > Home > ILE Home > Th. List > ssind | GIF version |
Description: A deduction showing that a subclass of two classes is a subclass of their intersection. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) |
Ref | Expression |
---|---|
ssind.1 | ⊢ (𝜑 → 𝐴 ⊆ 𝐵) |
ssind.2 | ⊢ (𝜑 → 𝐴 ⊆ 𝐶) |
Ref | Expression |
---|---|
ssind | ⊢ (𝜑 → 𝐴 ⊆ (𝐵 ∩ 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssind.1 | . 2 ⊢ (𝜑 → 𝐴 ⊆ 𝐵) | |
2 | ssind.2 | . 2 ⊢ (𝜑 → 𝐴 ⊆ 𝐶) | |
3 | ssin 3188 | . . 3 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐴 ⊆ 𝐶) ↔ 𝐴 ⊆ (𝐵 ∩ 𝐶)) | |
4 | 3 | biimpi 118 | . 2 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐴 ⊆ 𝐶) → 𝐴 ⊆ (𝐵 ∩ 𝐶)) |
5 | 1, 2, 4 | syl2anc 403 | 1 ⊢ (𝜑 → 𝐴 ⊆ (𝐵 ∩ 𝐶)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 102 ∩ cin 2972 ⊆ wss 2973 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 |
This theorem depends on definitions: df-bi 115 df-tru 1287 df-nf 1390 df-sb 1686 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-v 2603 df-in 2979 df-ss 2986 |
This theorem is referenced by: (None) |
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