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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 4533 | . 2 ⊢ (〈〈𝐴, 𝐵〉, 𝐶〉 ∈ ((𝑅 × 𝑆) × 𝑇) → 〈𝐴, 𝐵〉 ∈ (𝑅 × 𝑆)) | |
2 | opelxp1 4533 | . 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 1463 〈cop 3496 × cxp 4497 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 681 ax-5 1406 ax-7 1407 ax-gen 1408 ax-ie1 1452 ax-ie2 1453 ax-8 1465 ax-10 1466 ax-11 1467 ax-i12 1468 ax-bndl 1469 ax-4 1470 ax-14 1475 ax-17 1489 ax-i9 1493 ax-ial 1497 ax-i5r 1498 ax-ext 2097 ax-sep 4006 ax-pow 4058 ax-pr 4091 |
This theorem depends on definitions: df-bi 116 df-3an 947 df-tru 1317 df-nf 1420 df-sb 1719 df-clab 2102 df-cleq 2108 df-clel 2111 df-nfc 2244 df-ral 2395 df-rex 2396 df-v 2659 df-un 3041 df-in 3043 df-ss 3050 df-pw 3478 df-sn 3499 df-pr 3500 df-op 3502 df-opab 3950 df-xp 4505 |
This theorem is referenced by: (None) |
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