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| Mirrors > Home > MPE Home > Th. List > ssfid | Structured version Visualization version GIF version | ||
| Description: A subset of a finite set is finite, deduction version of ssfi 9213. (Contributed by Glauco Siliprandi, 21-Nov-2020.) |
| Ref | Expression |
|---|---|
| ssfid.1 | ⊢ (𝜑 → 𝐴 ∈ Fin) |
| ssfid.2 | ⊢ (𝜑 → 𝐵 ⊆ 𝐴) |
| Ref | Expression |
|---|---|
| ssfid | ⊢ (𝜑 → 𝐵 ∈ Fin) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | ssfid.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ Fin) | |
| 2 | ssfid.2 | . 2 ⊢ (𝜑 → 𝐵 ⊆ 𝐴) | |
| 3 | ssfi 9213 | . 2 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ⊆ 𝐴) → 𝐵 ∈ Fin) | |
| 4 | 1, 2, 3 | syl2anc 584 | 1 ⊢ (𝜑 → 𝐵 ∈ Fin) |
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