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| Mirrors > Home > MPE Home > Th. List > opelxpd | Structured version Visualization version GIF version | ||
| Description: Ordered pair membership in a Cartesian product, deduction form. (Contributed by Glauco Siliprandi, 3-Mar-2021.) |
| Ref | Expression |
|---|---|
| opelxpd.1 | ⊢ (𝜑 → 𝐴 ∈ 𝐶) |
| opelxpd.2 | ⊢ (𝜑 → 𝐵 ∈ 𝐷) |
| Ref | Expression |
|---|---|
| opelxpd | ⊢ (𝜑 → 〈𝐴, 𝐵〉 ∈ (𝐶 × 𝐷)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | opelxpd.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ 𝐶) | |
| 2 | opelxpd.2 | . 2 ⊢ (𝜑 → 𝐵 ∈ 𝐷) | |
| 3 | opelxpi 5722 | . 2 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → 〈𝐴, 𝐵〉 ∈ (𝐶 × 𝐷)) | |
| 4 | 1, 2, 3 | syl2anc 584 | 1 ⊢ (𝜑 → 〈𝐴, 𝐵〉 ∈ (𝐶 × 𝐷)) |
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