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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 3357 | . . 3 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐴 ⊆ 𝐶) ↔ 𝐴 ⊆ (𝐵 ∩ 𝐶)) | |
4 | 3 | biimpi 120 | . 2 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐴 ⊆ 𝐶) → 𝐴 ⊆ (𝐵 ∩ 𝐶)) |
5 | 1, 2, 4 | syl2anc 411 | 1 ⊢ (𝜑 → 𝐴 ⊆ (𝐵 ∩ 𝐶)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ∩ cin 3128 ⊆ wss 3129 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-ext 2159 |
This theorem depends on definitions: df-bi 117 df-tru 1356 df-nf 1461 df-sb 1763 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-v 2739 df-in 3135 df-ss 3142 |
This theorem is referenced by: ntrin 13257 lmss 13379 |
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