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| Description: Ordered triple membership in a triple Cartesian product. (Contributed by Scott Fenton, 31-Jan-2025.) | 
| Ref | Expression | 
|---|---|
| otelxp | ⊢ (〈𝐴, 𝐵, 𝐶〉 ∈ ((𝐷 × 𝐸) × 𝐹) ↔ (𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝐸 ∧ 𝐶 ∈ 𝐹)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | opelxp 5720 | . . 3 ⊢ (〈〈𝐴, 𝐵〉, 𝐶〉 ∈ ((𝐷 × 𝐸) × 𝐹) ↔ (〈𝐴, 𝐵〉 ∈ (𝐷 × 𝐸) ∧ 𝐶 ∈ 𝐹)) | |
| 2 | opelxp 5720 | . . 3 ⊢ (〈𝐴, 𝐵〉 ∈ (𝐷 × 𝐸) ↔ (𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝐸)) | |
| 3 | 1, 2 | bianbi 627 | . 2 ⊢ (〈〈𝐴, 𝐵〉, 𝐶〉 ∈ ((𝐷 × 𝐸) × 𝐹) ↔ ((𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝐸) ∧ 𝐶 ∈ 𝐹)) | 
| 4 | df-ot 4634 | . . 3 ⊢ 〈𝐴, 𝐵, 𝐶〉 = 〈〈𝐴, 𝐵〉, 𝐶〉 | |
| 5 | 4 | eleq1i 2831 | . 2 ⊢ (〈𝐴, 𝐵, 𝐶〉 ∈ ((𝐷 × 𝐸) × 𝐹) ↔ 〈〈𝐴, 𝐵〉, 𝐶〉 ∈ ((𝐷 × 𝐸) × 𝐹)) | 
| 6 | df-3an 1088 | . 2 ⊢ ((𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝐸 ∧ 𝐶 ∈ 𝐹) ↔ ((𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝐸) ∧ 𝐶 ∈ 𝐹)) | |
| 7 | 3, 5, 6 | 3bitr4i 303 | 1 ⊢ (〈𝐴, 𝐵, 𝐶〉 ∈ ((𝐷 × 𝐸) × 𝐹) ↔ (𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝐸 ∧ 𝐶 ∈ 𝐹)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ↔ wb 206 ∧ wa 395 ∧ w3a 1086 ∈ wcel 2107 〈cop 4631 〈cotp 4633 × cxp 5682 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pr 5431 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4626 df-pr 4628 df-op 4632 df-ot 4634 df-opab 5205 df-xp 5690 | 
| This theorem is referenced by: frpoins3xp3g 8167 xpord3lem 8175 xpord3pred 8178 xpord3inddlem 8180 | 
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