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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 5072 | . 2 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → 〈𝐴, 𝐵〉 ∈ (𝐶 × 𝐷)) | |
4 | 1, 2, 3 | syl2anc 691 | 1 ⊢ (𝜑 → 〈𝐴, 𝐵〉 ∈ (𝐶 × 𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 1977 〈cop 4131 × cxp 5036 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-opab 4644 df-xp 5044 |
This theorem is referenced by: otel3xp 5077 txkgen 21265 cnheiborlem 22561 esum2dlem 29481 mblfinlem2 32617 ovolval4lem1 39539 ovolval5lem2 39543 av-numclwlk1lem2f 41522 |
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