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Mirrors > Home > MPE Home > Th. List > Mathboxes > ot2elxp | Structured version Visualization version GIF version |
Description: Ordered triple membership in a triple cross product. (Contributed by Scott Fenton, 21-Aug-2024.) |
Ref | Expression |
---|---|
ot2elxp | ⊢ (〈〈𝐴, 𝐵〉, 𝐶〉 ∈ ((𝐷 × 𝐸) × 𝐹) ↔ (𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝐸 ∧ 𝐶 ∈ 𝐹)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | opelxp 5627 | . . 3 ⊢ (〈𝐴, 𝐵〉 ∈ (𝐷 × 𝐸) ↔ (𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝐸)) | |
2 | 1 | anbi1i 623 | . 2 ⊢ ((〈𝐴, 𝐵〉 ∈ (𝐷 × 𝐸) ∧ 𝐶 ∈ 𝐹) ↔ ((𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝐸) ∧ 𝐶 ∈ 𝐹)) |
3 | opelxp 5627 | . 2 ⊢ (〈〈𝐴, 𝐵〉, 𝐶〉 ∈ ((𝐷 × 𝐸) × 𝐹) ↔ (〈𝐴, 𝐵〉 ∈ (𝐷 × 𝐸) ∧ 𝐶 ∈ 𝐹)) | |
4 | df-3an 1087 | . 2 ⊢ ((𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝐸 ∧ 𝐶 ∈ 𝐹) ↔ ((𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝐸) ∧ 𝐶 ∈ 𝐹)) | |
5 | 2, 3, 4 | 3bitr4i 302 | 1 ⊢ (〈〈𝐴, 𝐵〉, 𝐶〉 ∈ ((𝐷 × 𝐸) × 𝐹) ↔ (𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝐸 ∧ 𝐶 ∈ 𝐹)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 395 ∧ w3a 1085 ∈ wcel 2101 〈cop 4570 × cxp 5589 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2103 ax-9 2111 ax-ext 2704 ax-sep 5226 ax-nul 5233 ax-pr 5355 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3an 1087 df-tru 1540 df-fal 1550 df-ex 1778 df-sb 2063 df-clab 2711 df-cleq 2725 df-clel 2811 df-ral 3060 df-rex 3069 df-rab 3224 df-v 3436 df-dif 3892 df-un 3894 df-in 3896 df-ss 3906 df-nul 4260 df-if 4463 df-sn 4565 df-pr 4567 df-op 4571 df-opab 5140 df-xp 5597 |
This theorem is referenced by: frpoins3xp3g 33816 xpord3lem 33823 xpord3ind 33828 |
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