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Theorem otexg 4081
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 3476 . . 3 𝐴, 𝐵, 𝐶⟩ = ⟨⟨𝐴, 𝐵⟩, 𝐶
2 opexg 4079 . . . 4 ((𝐴𝑈𝐵𝑉) → ⟨𝐴, 𝐵⟩ ∈ V)
3 opexg 4079 . . . 4 ((⟨𝐴, 𝐵⟩ ∈ V ∧ 𝐶𝑊) → ⟨⟨𝐴, 𝐵⟩, 𝐶⟩ ∈ V)
42, 3sylan 278 . . 3 (((𝐴𝑈𝐵𝑉) ∧ 𝐶𝑊) → ⟨⟨𝐴, 𝐵⟩, 𝐶⟩ ∈ V)
51, 4syl5eqel 2181 . 2 (((𝐴𝑈𝐵𝑉) ∧ 𝐶𝑊) → ⟨𝐴, 𝐵, 𝐶⟩ ∈ V)
653impa 1141 1 ((𝐴𝑈𝐵𝑉𝐶𝑊) → ⟨𝐴, 𝐵, 𝐶⟩ ∈ V)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  w3a 927  wcel 1445  Vcvv 2633  cop 3469  cotp 3470
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-14 1457  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077  ax-sep 3978  ax-pow 4030  ax-pr 4060
This theorem depends on definitions:  df-bi 116  df-3an 929  df-tru 1299  df-nf 1402  df-sb 1700  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-v 2635  df-un 3017  df-in 3019  df-ss 3026  df-pw 3451  df-sn 3472  df-pr 3473  df-op 3475  df-ot 3476
This theorem is referenced by:  euotd  4105
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