Theorem wunot 10761

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 4640	. 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 10760	. . 3 ⊢ (𝜑 → ⟨𝐴, 𝐵⟩ ∈ 𝑈)
6		wunot.3	. . 3 ⊢ (𝜑 → 𝐶 ∈ 𝑈)
7	2, 5, 6	wunop 10760	. 2 ⊢ (𝜑 → ⟨⟨𝐴, 𝐵⟩, 𝐶⟩ ∈ 𝑈)
8	1, 7	eqeltrid 2843	1 ⊢ (𝜑 → ⟨𝐴, 𝐵, 𝐶⟩ ∈ 𝑈)

Syntax hints:   → wi 4   ∈ wcel 2106  ⟨cop 4637  ⟨cotp 4639  WUnicwun 10738

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-ral 3060  df-rex 3069  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-ot 4640  df-uni 4913  df-tr 5266  df-wun 10740

This theorem is referenced by: (None)