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Mirrors > Home > ILE Home > Th. List > otelxp1 | GIF version |
Description: The first member of an ordered triple of classes in a cross product belongs to first cross 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 4674 | . 2 ⊢ (〈〈𝐴, 𝐵〉, 𝐶〉 ∈ ((𝑅 × 𝑆) × 𝑇) → 〈𝐴, 𝐵〉 ∈ (𝑅 × 𝑆)) | |
2 | opelxp1 4674 | . 2 ⊢ (〈𝐴, 𝐵〉 ∈ (𝑅 × 𝑆) → 𝐴 ∈ 𝑅) | |
3 | 1, 2 | syl 14 | 1 ⊢ (〈〈𝐴, 𝐵〉, 𝐶〉 ∈ ((𝑅 × 𝑆) × 𝑇) → 𝐴 ∈ 𝑅) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 2159 〈cop 3609 × cxp 4638 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 710 ax-5 1457 ax-7 1458 ax-gen 1459 ax-ie1 1503 ax-ie2 1504 ax-8 1514 ax-10 1515 ax-11 1516 ax-i12 1517 ax-bndl 1519 ax-4 1520 ax-17 1536 ax-i9 1540 ax-ial 1544 ax-i5r 1545 ax-14 2162 ax-ext 2170 ax-sep 4135 ax-pow 4188 ax-pr 4223 |
This theorem depends on definitions: df-bi 117 df-3an 981 df-tru 1366 df-nf 1471 df-sb 1773 df-clab 2175 df-cleq 2181 df-clel 2184 df-nfc 2320 df-ral 2472 df-rex 2473 df-v 2753 df-un 3147 df-in 3149 df-ss 3156 df-pw 3591 df-sn 3612 df-pr 3613 df-op 3615 df-opab 4079 df-xp 4646 |
This theorem is referenced by: (None) |
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