Theorem wunot 10763

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 4635	. 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 10762	. . 3 ⊢ (𝜑 → ⟨𝐴, 𝐵⟩ ∈ 𝑈)
6		wunot.3	. . 3 ⊢ (𝜑 → 𝐶 ∈ 𝑈)
7	2, 5, 6	wunop 10762	. 2 ⊢ (𝜑 → ⟨⟨𝐴, 𝐵⟩, 𝐶⟩ ∈ 𝑈)
8	1, 7	eqeltrid 2845	1 ⊢ (𝜑 → ⟨𝐴, 𝐵, 𝐶⟩ ∈ 𝑈)

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)