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Mirrors > Home > ILE Home > Th. List > otexg | GIF version |
Description: An ordered triple of sets is a set. (Contributed by Jim Kingdon, 19-Sep-2018.) |
Ref | Expression |
---|---|
otexg | ⊢ ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → 〈𝐴, 𝐵, 𝐶〉 ∈ V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-ot 3537 | . . 3 ⊢ 〈𝐴, 𝐵, 𝐶〉 = 〈〈𝐴, 𝐵〉, 𝐶〉 | |
2 | opexg 4150 | . . . 4 ⊢ ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉) → 〈𝐴, 𝐵〉 ∈ V) | |
3 | opexg 4150 | . . . 4 ⊢ ((〈𝐴, 𝐵〉 ∈ V ∧ 𝐶 ∈ 𝑊) → 〈〈𝐴, 𝐵〉, 𝐶〉 ∈ V) | |
4 | 2, 3 | sylan 281 | . . 3 ⊢ (((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉) ∧ 𝐶 ∈ 𝑊) → 〈〈𝐴, 𝐵〉, 𝐶〉 ∈ V) |
5 | 1, 4 | eqeltrid 2226 | . 2 ⊢ (((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉) ∧ 𝐶 ∈ 𝑊) → 〈𝐴, 𝐵, 𝐶〉 ∈ V) |
6 | 5 | 3impa 1176 | 1 ⊢ ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → 〈𝐴, 𝐵, 𝐶〉 ∈ V) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ∧ w3a 962 ∈ wcel 1480 Vcvv 2686 〈cop 3530 〈cotp 3531 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-v 2688 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-ot 3537 |
This theorem is referenced by: euotd 4176 |
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