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Mirrors > Home > ILE Home > Th. List > sylan9ss | GIF version |
Description: A subclass transitivity deduction. (Contributed by NM, 27-Sep-2004.) (Proof shortened by Andrew Salmon, 14-Jun-2011.) |
Ref | Expression |
---|---|
sylan9ss.1 | ⊢ (𝜑 → 𝐴 ⊆ 𝐵) |
sylan9ss.2 | ⊢ (𝜓 → 𝐵 ⊆ 𝐶) |
Ref | Expression |
---|---|
sylan9ss | ⊢ ((𝜑 ∧ 𝜓) → 𝐴 ⊆ 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sylan9ss.1 | . 2 ⊢ (𝜑 → 𝐴 ⊆ 𝐵) | |
2 | sylan9ss.2 | . 2 ⊢ (𝜓 → 𝐵 ⊆ 𝐶) | |
3 | sstr 3178 | . 2 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐶) → 𝐴 ⊆ 𝐶) | |
4 | 1, 2, 3 | syl2an 289 | 1 ⊢ ((𝜑 ∧ 𝜓) → 𝐴 ⊆ 𝐶) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ⊆ wss 3144 |
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-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-11 1517 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-ext 2171 |
This theorem depends on definitions: df-bi 117 df-nf 1472 df-sb 1774 df-clab 2176 df-cleq 2182 df-clel 2185 df-in 3150 df-ss 3157 |
This theorem is referenced by: sylan9ssr 3184 unss12 3322 ss2in 3378 relrelss 5170 funssxp 5400 |
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