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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 3377 | . . 3 ⊢ 〈A, B, 𝐶〉 = 〈〈A, B〉, 𝐶〉 | |
2 | opexg 3955 | . . . 4 ⊢ ((A ∈ 𝑈 ∧ B ∈ 𝑉) → 〈A, B〉 ∈ V) | |
3 | opexg 3955 | . . . 4 ⊢ ((〈A, B〉 ∈ V ∧ 𝐶 ∈ 𝑊) → 〈〈A, B〉, 𝐶〉 ∈ V) | |
4 | 2, 3 | sylan 267 | . . 3 ⊢ (((A ∈ 𝑈 ∧ B ∈ 𝑉) ∧ 𝐶 ∈ 𝑊) → 〈〈A, B〉, 𝐶〉 ∈ V) |
5 | 1, 4 | syl5eqel 2121 | . 2 ⊢ (((A ∈ 𝑈 ∧ B ∈ 𝑉) ∧ 𝐶 ∈ 𝑊) → 〈A, B, 𝐶〉 ∈ V) |
6 | 5 | 3impa 1098 | 1 ⊢ ((A ∈ 𝑈 ∧ B ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → 〈A, B, 𝐶〉 ∈ V) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 ∧ w3a 884 ∈ wcel 1390 Vcvv 2551 〈cop 3370 〈cotp 3371 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-14 1402 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 ax-sep 3866 ax-pow 3918 ax-pr 3935 |
This theorem depends on definitions: df-bi 110 df-3an 886 df-tru 1245 df-nf 1347 df-sb 1643 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-v 2553 df-un 2916 df-in 2918 df-ss 2925 df-pw 3353 df-sn 3373 df-pr 3374 df-op 3376 df-ot 3377 |
This theorem is referenced by: euotd 3982 |
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