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Mirrors > Home > ILE Home > Th. List > ssrdv | 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 1861 | . 2 ⊢ (𝜑 → ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵)) |
3 | dfss2 3127 | . 2 ⊢ (𝐴 ⊆ 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵)) | |
4 | 2, 3 | sylibr 133 | 1 ⊢ (𝜑 → 𝐴 ⊆ 𝐵) |
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