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Theorem wunot 10763
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 4635 . 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 10762 . . 3 (𝜑 → ⟨𝐴, 𝐵⟩ ∈ 𝑈)
6 wunot.3 . . 3 (𝜑𝐶𝑈)
72, 5, 6wunop 10762 . 2 (𝜑 → ⟨⟨𝐴, 𝐵⟩, 𝐶⟩ ∈ 𝑈)
81, 7eqeltrid 2845 1 (𝜑 → ⟨𝐴, 𝐵, 𝐶⟩ ∈ 𝑈)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2108  cop 4632  cotp 4634  WUnicwun 10740
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3062  df-rex 3071  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-ot 4635  df-uni 4908  df-tr 5260  df-wun 10742
This theorem is referenced by: (None)
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