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Theorem wunot 10143
Description: A weak universe is closed under ordered triples. (Contributed by Mario Carneiro, 2-Jan-2017.)
Hypotheses
Ref Expression
wun0.1 (𝜑𝑈 ∈ WUni)
wunop.2 (𝜑𝐴𝑈)
wunop.3 (𝜑𝐵𝑈)
wunot.3 (𝜑𝐶𝑈)
Assertion
Ref Expression
wunot (𝜑 → ⟨𝐴, 𝐵, 𝐶⟩ ∈ 𝑈)

Proof of Theorem wunot
StepHypRef Expression
1 df-ot 4559 . 2 𝐴, 𝐵, 𝐶⟩ = ⟨⟨𝐴, 𝐵⟩, 𝐶
2 wun0.1 . . 3 (𝜑𝑈 ∈ WUni)
3 wunop.2 . . . 4 (𝜑𝐴𝑈)
4 wunop.3 . . . 4 (𝜑𝐵𝑈)
52, 3, 4wunop 10142 . . 3 (𝜑 → ⟨𝐴, 𝐵⟩ ∈ 𝑈)
6 wunot.3 . . 3 (𝜑𝐶𝑈)
72, 5, 6wunop 10142 . 2 (𝜑 → ⟨⟨𝐴, 𝐵⟩, 𝐶⟩ ∈ 𝑈)
81, 7eqeltrid 2920 1 (𝜑 → ⟨𝐴, 𝐵, 𝐶⟩ ∈ 𝑈)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2115  cop 4556  cotp 4558  WUnicwun 10120
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-v 3482  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-sn 4551  df-pr 4553  df-op 4557  df-ot 4559  df-uni 4825  df-tr 5159  df-wun 10122
This theorem is referenced by: (None)
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