Theorem wunot 10652

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 4594	. 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 10651	. . 3 ⊢ (𝜑 → ⟨𝐴, 𝐵⟩ ∈ 𝑈)
6		wunot.3	. . 3 ⊢ (𝜑 → 𝐶 ∈ 𝑈)
7	2, 5, 6	wunop 10651	. 2 ⊢ (𝜑 → ⟨⟨𝐴, 𝐵⟩, 𝐶⟩ ∈ 𝑈)
8	1, 7	eqeltrid 2832	1 ⊢ (𝜑 → ⟨𝐴, 𝐵, 𝐶⟩ ∈ 𝑈)

Syntax hints:   → wi 4   ∈ wcel 2109  ⟨cop 4591  ⟨cotp 4593  WUnicwun 10629

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 2008  ax-8 2111  ax-9 2119  ax-ext 2701

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-ne 2926  df-ral 3045  df-rex 3054  df-v 3446  df-dif 3914  df-un 3916  df-ss 3928  df-nul 4293  df-if 4485  df-sn 4586  df-pr 4588  df-op 4592  df-ot 4594  df-uni 4868  df-tr 5210  df-wun 10631

This theorem is referenced by: (None)