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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 3591 | . . 3 ⊢ 〈𝐴, 𝐵, 𝐶〉 = 〈〈𝐴, 𝐵〉, 𝐶〉 | |
2 | opexg 4211 | . . . 4 ⊢ ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉) → 〈𝐴, 𝐵〉 ∈ V) | |
3 | opexg 4211 | . . . 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 2257 | . 2 ⊢ (((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉) ∧ 𝐶 ∈ 𝑊) → 〈𝐴, 𝐵, 𝐶〉 ∈ V) |
6 | 5 | 3impa 1189 | 1 ⊢ ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → 〈𝐴, 𝐵, 𝐶〉 ∈ V) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ∧ w3a 973 ∈ wcel 2141 Vcvv 2730 〈cop 3584 〈cotp 3585 |
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 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-14 2144 ax-ext 2152 ax-sep 4105 ax-pow 4158 ax-pr 4192 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-nf 1454 df-sb 1756 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-v 2732 df-un 3125 df-in 3127 df-ss 3134 df-pw 3566 df-sn 3587 df-pr 3588 df-op 3590 df-ot 3591 |
This theorem is referenced by: euotd 4237 |
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