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| Mirrors > Home > MPE Home > Th. List > sstrd | Structured version Visualization version GIF version | ||
| Description: Subclass transitivity deduction. (Contributed by NM, 2-Jun-2004.) |
| Ref | Expression |
|---|---|
| sstrd.1 | ⊢ (𝜑 → 𝐴 ⊆ 𝐵) |
| sstrd.2 | ⊢ (𝜑 → 𝐵 ⊆ 𝐶) |
| Ref | Expression |
|---|---|
| sstrd | ⊢ (𝜑 → 𝐴 ⊆ 𝐶) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | sstrd.1 | . 2 ⊢ (𝜑 → 𝐴 ⊆ 𝐵) | |
| 2 | sstrd.2 | . 2 ⊢ (𝜑 → 𝐵 ⊆ 𝐶) | |
| 3 | sstr 3992 | . 2 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐶) → 𝐴 ⊆ 𝐶) | |
| 4 | 1, 2, 3 | syl2anc 584 | 1 ⊢ (𝜑 → 𝐴 ⊆ 𝐶) |
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