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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 5564 | . . 3 ⊢ (〈𝐴, 𝐵〉 ∈ (𝐷 × 𝐸) ↔ (𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝐸)) | |
2 | 1 | anbi1i 626 | . 2 ⊢ ((〈𝐴, 𝐵〉 ∈ (𝐷 × 𝐸) ∧ 𝐶 ∈ 𝐹) ↔ ((𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝐸) ∧ 𝐶 ∈ 𝐹)) |
3 | opelxp 5564 | . 2 ⊢ (〈〈𝐴, 𝐵〉, 𝐶〉 ∈ ((𝐷 × 𝐸) × 𝐹) ↔ (〈𝐴, 𝐵〉 ∈ (𝐷 × 𝐸) ∧ 𝐶 ∈ 𝐹)) | |
4 | df-3an 1086 | . 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 1084 ∈ wcel 2111 〈cop 4531 × cxp 5526 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-ext 2729 ax-sep 5173 ax-nul 5180 ax-pr 5302 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-fal 1551 df-ex 1782 df-sb 2070 df-clab 2736 df-cleq 2750 df-clel 2830 df-ral 3075 df-rex 3076 df-v 3411 df-dif 3863 df-un 3865 df-nul 4228 df-if 4424 df-sn 4526 df-pr 4528 df-op 4532 df-opab 5099 df-xp 5534 |
This theorem is referenced by: frpoins3xp3g 33345 xpord3lem 33362 xpord3ind 33367 |
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