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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 4566 | . . . 4 ⊢ 〈𝐴, 𝐵, 𝐶〉 = 〈〈𝐴, 𝐵〉, 𝐶〉 | |
2 | 3simpa 1140 | . . . . . 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 235 | . . . . 5 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) → 〈𝐴, 𝐵〉 ∈ (𝑋 × 𝑌)) |
5 | simp3 1130 | . . . . 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 2914 | . . 3 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) → 〈𝐴, 𝐵, 𝐶〉 ∈ ((𝑋 × 𝑌) × 𝑍)) |
8 | eleq1 2897 | . . 3 ⊢ (𝑇 = 〈𝐴, 𝐵, 𝐶〉 → (𝑇 ∈ ((𝑋 × 𝑌) × 𝑍) ↔ 〈𝐴, 𝐵, 𝐶〉 ∈ ((𝑋 × 𝑌) × 𝑍))) | |
9 | 7, 8 | syl5ibr 247 | . 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 1079 = wceq 1528 ∈ wcel 2105 〈cop 4563 〈cotp 4565 × cxp 5546 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pr 5320 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-ot 4566 df-opab 5120 df-xp 5554 |
This theorem is referenced by: (None) |
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