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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 3047 | . 2 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐶) → 𝐴 ⊆ 𝐶) | |
4 | 1, 2, 3 | syl2an 284 | 1 ⊢ ((𝜑 ∧ 𝜓) → 𝐴 ⊆ 𝐶) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ⊆ wss 3013 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-5 1388 ax-7 1389 ax-gen 1390 ax-ie1 1434 ax-ie2 1435 ax-8 1447 ax-11 1449 ax-4 1452 ax-17 1471 ax-i9 1475 ax-ial 1479 ax-i5r 1480 ax-ext 2077 |
This theorem depends on definitions: df-bi 116 df-nf 1402 df-sb 1700 df-clab 2082 df-cleq 2088 df-clel 2091 df-in 3019 df-ss 3026 |
This theorem is referenced by: sylan9ssr 3053 unss12 3187 ss2in 3243 relrelss 4991 funssxp 5215 |
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