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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 4569 | . . . 4 ⊢ 〈𝐴, 𝐵, 𝐶〉 = 〈〈𝐴, 𝐵〉, 𝐶〉 | |
2 | 3simpa 1143 | . . . . . 6 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) → (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) | |
3 | opelxp 5584 | . . . . . 6 ⊢ (〈𝐴, 𝐵〉 ∈ (𝑋 × 𝑌) ↔ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) | |
4 | 2, 3 | sylibr 236 | . . . . 5 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) → 〈𝐴, 𝐵〉 ∈ (𝑋 × 𝑌)) |
5 | simp3 1133 | . . . . 5 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) → 𝐶 ∈ 𝑍) | |
6 | 4, 5 | opelxpd 5586 | . . . 4 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) → 〈〈𝐴, 𝐵〉, 𝐶〉 ∈ ((𝑋 × 𝑌) × 𝑍)) |
7 | 1, 6 | eqeltrid 2916 | . . 3 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) → 〈𝐴, 𝐵, 𝐶〉 ∈ ((𝑋 × 𝑌) × 𝑍)) |
8 | eleq1 2899 | . . 3 ⊢ (𝑇 = 〈𝐴, 𝐵, 𝐶〉 → (𝑇 ∈ ((𝑋 × 𝑌) × 𝑍) ↔ 〈𝐴, 𝐵, 𝐶〉 ∈ ((𝑋 × 𝑌) × 𝑍))) | |
9 | 7, 8 | syl5ibr 248 | . 2 ⊢ (𝑇 = 〈𝐴, 𝐵, 𝐶〉 → ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) → 𝑇 ∈ ((𝑋 × 𝑌) × 𝑍))) |
10 | 9 | imp 409 | 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 398 ∧ w3a 1082 = wceq 1536 ∈ wcel 2113 〈cop 4566 〈cotp 4568 × cxp 5546 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 ax-sep 5196 ax-nul 5203 ax-pr 5323 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1084 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ral 3142 df-rex 3143 df-rab 3146 df-v 3493 df-dif 3932 df-un 3934 df-in 3936 df-ss 3945 df-nul 4285 df-if 4461 df-sn 4561 df-pr 4563 df-op 4567 df-ot 4569 df-opab 5122 df-xp 5554 |
This theorem is referenced by: (None) |
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