Theorem wunot 9861

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 4407	. 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 9860	. . 3 ⊢ (𝜑 → ⟨𝐴, 𝐵⟩ ∈ 𝑈)
6		wunot.3	. . 3 ⊢ (𝜑 → 𝐶 ∈ 𝑈)
7	2, 5, 6	wunop 9860	. 2 ⊢ (𝜑 → ⟨⟨𝐴, 𝐵⟩, 𝐶⟩ ∈ 𝑈)
8	1, 7	syl5eqel 2911	1 ⊢ (𝜑 → ⟨𝐴, 𝐵, 𝐶⟩ ∈ 𝑈)

Syntax hints:   → wi 4   ∈ wcel 2166  ⟨cop 4404  ⟨cotp 4406  WUnicwun 9838

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2804

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-clab 2813  df-cleq 2819  df-clel 2822  df-nfc 2959  df-ne 3001  df-ral 3123  df-rex 3124  df-v 3417  df-dif 3802  df-un 3804  df-in 3806  df-ss 3813  df-nul 4146  df-if 4308  df-sn 4399  df-pr 4401  df-op 4405  df-ot 4407  df-uni 4660  df-tr 4977  df-wun 9840

This theorem is referenced by: (None)