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Theorem otexg 4213
Description: An ordered triple of sets is a set. (Contributed by Jim Kingdon, 19-Sep-2018.)
Assertion
Ref Expression
otexg ((𝐴𝑈𝐵𝑉𝐶𝑊) → ⟨𝐴, 𝐵, 𝐶⟩ ∈ V)

Proof of Theorem otexg
StepHypRef Expression
1 df-ot 3591 . . 3 𝐴, 𝐵, 𝐶⟩ = ⟨⟨𝐴, 𝐵⟩, 𝐶
2 opexg 4211 . . . 4 ((𝐴𝑈𝐵𝑉) → ⟨𝐴, 𝐵⟩ ∈ V)
3 opexg 4211 . . . 4 ((⟨𝐴, 𝐵⟩ ∈ V ∧ 𝐶𝑊) → ⟨⟨𝐴, 𝐵⟩, 𝐶⟩ ∈ V)
42, 3sylan 281 . . 3 (((𝐴𝑈𝐵𝑉) ∧ 𝐶𝑊) → ⟨⟨𝐴, 𝐵⟩, 𝐶⟩ ∈ V)
51, 4eqeltrid 2257 . 2 (((𝐴𝑈𝐵𝑉) ∧ 𝐶𝑊) → ⟨𝐴, 𝐵, 𝐶⟩ ∈ V)
653impa 1189 1 ((𝐴𝑈𝐵𝑉𝐶𝑊) → ⟨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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