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Mirrors > Home > MPE Home > Th. List > otel3xp | Structured version Visualization version GIF version |
Description: An ordered triple is an element of a doubled Cartesian product. (Contributed by Alexander van der Vekens, 26-Feb-2018.) |
Ref | Expression |
---|---|
otel3xp | ⊢ ((𝑇 = 〈𝐴, 𝐵, 𝐶〉 ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍)) → 𝑇 ∈ ((𝑋 × 𝑌) × 𝑍)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-ot 4570 | . . . 4 ⊢ 〈𝐴, 𝐵, 𝐶〉 = 〈〈𝐴, 𝐵〉, 𝐶〉 | |
2 | 3simpa 1147 | . . . . . 6 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) → (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) | |
3 | opelxp 5625 | . . . . . 6 ⊢ (〈𝐴, 𝐵〉 ∈ (𝑋 × 𝑌) ↔ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) | |
4 | 2, 3 | sylibr 233 | . . . . 5 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) → 〈𝐴, 𝐵〉 ∈ (𝑋 × 𝑌)) |
5 | simp3 1137 | . . . . 5 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) → 𝐶 ∈ 𝑍) | |
6 | 4, 5 | opelxpd 5627 | . . . 4 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) → 〈〈𝐴, 𝐵〉, 𝐶〉 ∈ ((𝑋 × 𝑌) × 𝑍)) |
7 | 1, 6 | eqeltrid 2843 | . . 3 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) → 〈𝐴, 𝐵, 𝐶〉 ∈ ((𝑋 × 𝑌) × 𝑍)) |
8 | eleq1 2826 | . . 3 ⊢ (𝑇 = 〈𝐴, 𝐵, 𝐶〉 → (𝑇 ∈ ((𝑋 × 𝑌) × 𝑍) ↔ 〈𝐴, 𝐵, 𝐶〉 ∈ ((𝑋 × 𝑌) × 𝑍))) | |
9 | 7, 8 | syl5ibr 245 | . 2 ⊢ (𝑇 = 〈𝐴, 𝐵, 𝐶〉 → ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) → 𝑇 ∈ ((𝑋 × 𝑌) × 𝑍))) |
10 | 9 | imp 407 | 1 ⊢ ((𝑇 = 〈𝐴, 𝐵, 𝐶〉 ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍)) → 𝑇 ∈ ((𝑋 × 𝑌) × 𝑍)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∧ w3a 1086 = wceq 1539 ∈ wcel 2106 〈cop 4567 〈cotp 4569 × cxp 5587 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2709 ax-sep 5223 ax-nul 5230 ax-pr 5352 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-sb 2068 df-clab 2716 df-cleq 2730 df-clel 2816 df-ral 3069 df-rex 3070 df-rab 3073 df-v 3434 df-dif 3890 df-un 3892 df-nul 4257 df-if 4460 df-sn 4562 df-pr 4564 df-op 4568 df-ot 4570 df-opab 5137 df-xp 5595 |
This theorem is referenced by: (None) |
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