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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 4186 | . . . 4 ⊢ 〈𝐴, 𝐵, 𝐶〉 = 〈〈𝐴, 𝐵〉, 𝐶〉 | |
2 | 3simpa 1058 | . . . . . 6 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) → (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) | |
3 | opelxp 5146 | . . . . . 6 ⊢ (〈𝐴, 𝐵〉 ∈ (𝑋 × 𝑌) ↔ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) | |
4 | 2, 3 | sylibr 224 | . . . . 5 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) → 〈𝐴, 𝐵〉 ∈ (𝑋 × 𝑌)) |
5 | simp3 1063 | . . . . 5 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) → 𝐶 ∈ 𝑍) | |
6 | 4, 5 | opelxpd 5149 | . . . 4 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) → 〈〈𝐴, 𝐵〉, 𝐶〉 ∈ ((𝑋 × 𝑌) × 𝑍)) |
7 | 1, 6 | syl5eqel 2705 | . . 3 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) → 〈𝐴, 𝐵, 𝐶〉 ∈ ((𝑋 × 𝑌) × 𝑍)) |
8 | eleq1 2689 | . . 3 ⊢ (𝑇 = 〈𝐴, 𝐵, 𝐶〉 → (𝑇 ∈ ((𝑋 × 𝑌) × 𝑍) ↔ 〈𝐴, 𝐵, 𝐶〉 ∈ ((𝑋 × 𝑌) × 𝑍))) | |
9 | 7, 8 | syl5ibr 236 | . 2 ⊢ (𝑇 = 〈𝐴, 𝐵, 𝐶〉 → ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) → 𝑇 ∈ ((𝑋 × 𝑌) × 𝑍))) |
10 | 9 | imp 445 | 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 384 ∧ w3a 1037 = wceq 1483 ∈ wcel 1990 〈cop 4183 〈cotp 4185 × cxp 5112 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-ot 4186 df-opab 4713 df-xp 5120 |
This theorem is referenced by: (None) |
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