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Theorem wunot 10667
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 4599 . 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 10666 . . 3 (𝜑 → ⟨𝐴, 𝐵⟩ ∈ 𝑈)
6 wunot.3 . . 3 (𝜑𝐶𝑈)
72, 5, 6wunop 10666 . 2 (𝜑 → ⟨⟨𝐴, 𝐵⟩, 𝐶⟩ ∈ 𝑈)
81, 7eqeltrid 2838 1 (𝜑 → ⟨𝐴, 𝐵, 𝐶⟩ ∈ 𝑈)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2107  cop 4596  cotp 4598  WUnicwun 10644
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2941  df-ral 3062  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-sn 4591  df-pr 4593  df-op 4597  df-ot 4599  df-uni 4870  df-tr 5227  df-wun 10646
This theorem is referenced by: (None)
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