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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 5601 | . . 3 ⊢ (〈𝐴, 𝐵〉 ∈ (𝐷 × 𝐸) ↔ (𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝐸)) | |
2 | 1 | anbi1i 627 | . 2 ⊢ ((〈𝐴, 𝐵〉 ∈ (𝐷 × 𝐸) ∧ 𝐶 ∈ 𝐹) ↔ ((𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝐸) ∧ 𝐶 ∈ 𝐹)) |
3 | opelxp 5601 | . 2 ⊢ (〈〈𝐴, 𝐵〉, 𝐶〉 ∈ ((𝐷 × 𝐸) × 𝐹) ↔ (〈𝐴, 𝐵〉 ∈ (𝐷 × 𝐸) ∧ 𝐶 ∈ 𝐹)) | |
4 | df-3an 1091 | . 2 ⊢ ((𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝐸 ∧ 𝐶 ∈ 𝐹) ↔ ((𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝐸) ∧ 𝐶 ∈ 𝐹)) | |
5 | 2, 3, 4 | 3bitr4i 306 | 1 ⊢ (〈〈𝐴, 𝐵〉, 𝐶〉 ∈ ((𝐷 × 𝐸) × 𝐹) ↔ (𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝐸 ∧ 𝐶 ∈ 𝐹)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 ∧ wa 399 ∧ w3a 1089 ∈ wcel 2111 〈cop 4561 × cxp 5563 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2113 ax-9 2121 ax-ext 2709 ax-sep 5206 ax-nul 5213 ax-pr 5336 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-sb 2072 df-clab 2716 df-cleq 2730 df-clel 2817 df-ral 3067 df-rex 3068 df-rab 3071 df-v 3422 df-dif 3883 df-un 3885 df-nul 4252 df-if 4454 df-sn 4556 df-pr 4558 df-op 4562 df-opab 5130 df-xp 5571 |
This theorem is referenced by: frpoins3xp3g 33551 xpord3lem 33558 xpord3ind 33563 |
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