Theorem wuntp 10730

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

Hypotheses

Ref	Expression
wununi.1	⊢ (𝜑 → 𝑈 ∈ WUni)
wununi.2	⊢ (𝜑 → 𝐴 ∈ 𝑈)
wunpr.3	⊢ (𝜑 → 𝐵 ∈ 𝑈)
wuntp.3	⊢ (𝜑 → 𝐶 ∈ 𝑈)

Assertion

Ref	Expression
wuntp	⊢ (𝜑 → {𝐴, 𝐵, 𝐶} ∈ 𝑈)

Proof of Theorem wuntp

Step	Hyp	Ref	Expression
1		tpass 4733	. 2 ⊢ {𝐴, 𝐵, 𝐶} = ({𝐴} ∪ {𝐵, 𝐶})
2		wununi.1	. . 3 ⊢ (𝜑 → 𝑈 ∈ WUni)
3		dfsn2 4619	. . . 4 ⊢ {𝐴} = {𝐴, 𝐴}
4		wununi.2	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ 𝑈)
5	2, 4, 4	wunpr 10728	. . . 4 ⊢ (𝜑 → {𝐴, 𝐴} ∈ 𝑈)
6	3, 5	eqeltrid 2839	. . 3 ⊢ (𝜑 → {𝐴} ∈ 𝑈)
7		wunpr.3	. . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝑈)
8		wuntp.3	. . . 4 ⊢ (𝜑 → 𝐶 ∈ 𝑈)
9	2, 7, 8	wunpr 10728	. . 3 ⊢ (𝜑 → {𝐵, 𝐶} ∈ 𝑈)
10	2, 6, 9	wunun 10729	. 2 ⊢ (𝜑 → ({𝐴} ∪ {𝐵, 𝐶}) ∈ 𝑈)
11	1, 10	eqeltrid 2839	1 ⊢ (𝜑 → {𝐴, 𝐵, 𝐶} ∈ 𝑈)

Syntax hints:   → wi 4   ∈ wcel 2109   ∪ cun 3929  {csn 4606  {cpr 4608  {ctp 4610  WUnicwun 10719

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 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-ex 1780  df-sb 2066  df-clab 2715  df-cleq 2728  df-clel 2810  df-ne 2934  df-ral 3053  df-rex 3062  df-v 3466  df-un 3936  df-ss 3948  df-sn 4607  df-pr 4609  df-tp 4611  df-uni 4889  df-tr 5235  df-wun 10721

This theorem is referenced by:  catcfuccl  18136  catcxpccl  18224