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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 3153 | . . 3 ⊢ ((A ⊆ B ∧ A ⊆ 𝐶) ↔ A ⊆ (B ∩ 𝐶)) | |
4 | 3 | biimpi 113 | . 2 ⊢ ((A ⊆ B ∧ A ⊆ 𝐶) → A ⊆ (B ∩ 𝐶)) |
5 | 1, 2, 4 | syl2anc 391 | 1 ⊢ (φ → A ⊆ (B ∩ 𝐶)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 ∩ cin 2910 ⊆ wss 2911 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 |
This theorem depends on definitions: df-bi 110 df-tru 1245 df-nf 1347 df-sb 1643 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-v 2553 df-in 2918 df-ss 2925 |
This theorem is referenced by: (None) |
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