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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 2953 | . 2 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐶) → 𝐴 ⊆ 𝐶) | |
4 | 1, 2, 3 | syl2an 273 | 1 ⊢ ((𝜑 ∧ 𝜓) → 𝐴 ⊆ 𝐶) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 ⊆ wss 2917 |
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-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-11 1397 ax-4 1400 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 |
This theorem depends on definitions: df-bi 110 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-in 2924 df-ss 2931 |
This theorem is referenced by: sylan9ssr 2959 psstr 3049 sspsstr 3050 psssstr 3051 unss12 3115 ss2in 3164 relrelss 4844 funssxp 5060 |
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