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| Mirrors > Home > MPE Home > Th. List > ssrdv | Structured version Visualization version GIF version | ||
| Description: Deduction based on subclass definition. (Contributed by NM, 15-Nov-1995.) |
| Ref | Expression |
|---|---|
| ssrdv.1 | ⊢ (𝜑 → (𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵)) |
| Ref | Expression |
|---|---|
| ssrdv | ⊢ (𝜑 → 𝐴 ⊆ 𝐵) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | ssrdv.1 | . . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵)) | |
| 2 | 1 | alrimiv 1927 | . 2 ⊢ (𝜑 → ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵)) |
| 3 | df-ss 3968 | . 2 ⊢ (𝐴 ⊆ 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵)) | |
| 4 | 2, 3 | sylibr 234 | 1 ⊢ (𝜑 → 𝐴 ⊆ 𝐵) |
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