Theorem wunot 10717

Description: A weak universe is closed under ordered triples. (Contributed by Mario Carneiro, 2-Jan-2017.)

Hypotheses

Ref	Expression
wun0.1	⊢ (𝜑 → 𝑈 ∈ WUni)
wunop.2	⊢ (𝜑 → 𝐴 ∈ 𝑈)
wunop.3	⊢ (𝜑 → 𝐵 ∈ 𝑈)
wunot.3	⊢ (𝜑 → 𝐶 ∈ 𝑈)

Assertion

Ref	Expression
wunot	⊢ (𝜑 → ⟨𝐴, 𝐵, 𝐶⟩ ∈ 𝑈)

Proof of Theorem wunot

Step	Hyp	Ref	Expression
1		df-ot 4637	. 2 ⊢ ⟨𝐴, 𝐵, 𝐶⟩ = ⟨⟨𝐴, 𝐵⟩, 𝐶⟩
2		wun0.1	. . 3 ⊢ (𝜑 → 𝑈 ∈ WUni)
3		wunop.2	. . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑈)
4		wunop.3	. . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝑈)
5	2, 3, 4	wunop 10716	. . 3 ⊢ (𝜑 → ⟨𝐴, 𝐵⟩ ∈ 𝑈)
6		wunot.3	. . 3 ⊢ (𝜑 → 𝐶 ∈ 𝑈)
7	2, 5, 6	wunop 10716	. 2 ⊢ (𝜑 → ⟨⟨𝐴, 𝐵⟩, 𝐶⟩ ∈ 𝑈)
8	1, 7	eqeltrid 2837	1 ⊢ (𝜑 → ⟨𝐴, 𝐵, 𝐶⟩ ∈ 𝑈)

Syntax hints:   → wi 4   ∈ wcel 2106  ⟨cop 4634  ⟨cotp 4636  WUnicwun 10694

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-ne 2941  df-ral 3062  df-rex 3071  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-ot 4637  df-uni 4909  df-tr 5266  df-wun 10696

This theorem is referenced by: (None)