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Mirrors > Home > MPE Home > Th. List > otelxp1 | Structured version Visualization version GIF version |
Description: The first member of an ordered triple of classes in a Cartesian product belongs to first Cartesian product argument. (Contributed by NM, 28-May-2008.) |
Ref | Expression |
---|---|
otelxp1 | ⊢ (〈〈𝐴, 𝐵〉, 𝐶〉 ∈ ((𝑅 × 𝑆) × 𝑇) → 𝐴 ∈ 𝑅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | opelxp1 5673 | . 2 ⊢ (〈〈𝐴, 𝐵〉, 𝐶〉 ∈ ((𝑅 × 𝑆) × 𝑇) → 〈𝐴, 𝐵〉 ∈ (𝑅 × 𝑆)) | |
2 | opelxp1 5673 | . 2 ⊢ (〈𝐴, 𝐵〉 ∈ (𝑅 × 𝑆) → 𝐴 ∈ 𝑅) | |
3 | 1, 2 | syl 17 | 1 ⊢ (〈〈𝐴, 𝐵〉, 𝐶〉 ∈ ((𝑅 × 𝑆) × 𝑇) → 𝐴 ∈ 𝑅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2106 〈cop 4591 × cxp 5630 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2707 ax-sep 5255 ax-nul 5262 ax-pr 5383 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-sb 2068 df-clab 2714 df-cleq 2728 df-clel 2814 df-ral 3064 df-rex 3073 df-rab 3407 df-v 3446 df-dif 3912 df-un 3914 df-in 3916 df-ss 3926 df-nul 4282 df-if 4486 df-sn 4586 df-pr 4588 df-op 4592 df-opab 5167 df-xp 5638 |
This theorem is referenced by: (None) |
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