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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 3542 | . . 3 ⊢ 〈𝐴, 𝐵, 𝐶〉 = 〈〈𝐴, 𝐵〉, 𝐶〉 | |
2 | opexg 4158 | . . . 4 ⊢ ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉) → 〈𝐴, 𝐵〉 ∈ V) | |
3 | opexg 4158 | . . . 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 2227 | . 2 ⊢ (((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉) ∧ 𝐶 ∈ 𝑊) → 〈𝐴, 𝐵, 𝐶〉 ∈ V) |
6 | 5 | 3impa 1177 | 1 ⊢ ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → 〈𝐴, 𝐵, 𝐶〉 ∈ V) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ∧ w3a 963 ∈ wcel 1481 Vcvv 2689 〈cop 3535 〈cotp 3536 |
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 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-sep 4054 ax-pow 4106 ax-pr 4139 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-nf 1438 df-sb 1737 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-v 2691 df-un 3080 df-in 3082 df-ss 3089 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-ot 3542 |
This theorem is referenced by: euotd 4184 |
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