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Mirrors > Home > ILE Home > Th. List > ssini | GIF version |
Description: An inference showing that a subclass of two classes is a subclass of their intersection. (Contributed by NM, 24-Nov-2003.) |
Ref | Expression |
---|---|
ssini.1 | ⊢ 𝐴 ⊆ 𝐵 |
ssini.2 | ⊢ 𝐴 ⊆ 𝐶 |
Ref | Expression |
---|---|
ssini | ⊢ 𝐴 ⊆ (𝐵 ∩ 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssini.1 | . . 3 ⊢ 𝐴 ⊆ 𝐵 | |
2 | ssini.2 | . . 3 ⊢ 𝐴 ⊆ 𝐶 | |
3 | 1, 2 | pm3.2i 270 | . 2 ⊢ (𝐴 ⊆ 𝐵 ∧ 𝐴 ⊆ 𝐶) |
4 | ssin 3293 | . 2 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐴 ⊆ 𝐶) ↔ 𝐴 ⊆ (𝐵 ∩ 𝐶)) | |
5 | 3, 4 | mpbi 144 | 1 ⊢ 𝐴 ⊆ (𝐵 ∩ 𝐶) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 103 ∩ cin 3065 ⊆ wss 3066 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 |
This theorem depends on definitions: df-bi 116 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-v 2683 df-in 3072 df-ss 3079 |
This theorem is referenced by: inv1 3394 |
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