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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 3629 | . . 3 ⊢ 〈𝐴, 𝐵, 𝐶〉 = 〈〈𝐴, 𝐵〉, 𝐶〉 | |
2 | opexg 4258 | . . . 4 ⊢ ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉) → 〈𝐴, 𝐵〉 ∈ V) | |
3 | opexg 4258 | . . . 4 ⊢ ((〈𝐴, 𝐵〉 ∈ V ∧ 𝐶 ∈ 𝑊) → 〈〈𝐴, 𝐵〉, 𝐶〉 ∈ V) | |
4 | 2, 3 | sylan 283 | . . 3 ⊢ (((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉) ∧ 𝐶 ∈ 𝑊) → 〈〈𝐴, 𝐵〉, 𝐶〉 ∈ V) |
5 | 1, 4 | eqeltrid 2280 | . 2 ⊢ (((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉) ∧ 𝐶 ∈ 𝑊) → 〈𝐴, 𝐵, 𝐶〉 ∈ V) |
6 | 5 | 3impa 1196 | 1 ⊢ ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → 〈𝐴, 𝐵, 𝐶〉 ∈ V) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ∧ w3a 980 ∈ wcel 2164 Vcvv 2760 〈cop 3622 〈cotp 3623 |
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 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-14 2167 ax-ext 2175 ax-sep 4148 ax-pow 4204 ax-pr 4239 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-nf 1472 df-sb 1774 df-clab 2180 df-cleq 2186 df-clel 2189 df-nfc 2325 df-v 2762 df-un 3158 df-in 3160 df-ss 3167 df-pw 3604 df-sn 3625 df-pr 3626 df-op 3628 df-ot 3629 |
This theorem is referenced by: euotd 4284 |
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