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Mirrors > Home > ILE Home > Th. List > sylan9ssr | GIF version |
Description: A subclass transitivity deduction. (Contributed by NM, 27-Sep-2004.) |
Ref | Expression |
---|---|
sylan9ssr.1 | ⊢ (𝜑 → 𝐴 ⊆ 𝐵) |
sylan9ssr.2 | ⊢ (𝜓 → 𝐵 ⊆ 𝐶) |
Ref | Expression |
---|---|
sylan9ssr | ⊢ ((𝜓 ∧ 𝜑) → 𝐴 ⊆ 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sylan9ssr.1 | . . 3 ⊢ (𝜑 → 𝐴 ⊆ 𝐵) | |
2 | sylan9ssr.2 | . . 3 ⊢ (𝜓 → 𝐵 ⊆ 𝐶) | |
3 | 1, 2 | sylan9ss 3168 | . 2 ⊢ ((𝜑 ∧ 𝜓) → 𝐴 ⊆ 𝐶) |
4 | 3 | ancoms 268 | 1 ⊢ ((𝜓 ∧ 𝜑) → 𝐴 ⊆ 𝐶) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ⊆ wss 3129 |
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 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-11 1506 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-ext 2159 |
This theorem depends on definitions: df-bi 117 df-nf 1461 df-sb 1763 df-clab 2164 df-cleq 2170 df-clel 2173 df-in 3135 df-ss 3142 |
This theorem is referenced by: intssuni2m 3866 |
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