Theorem wuntp 10127

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 4682	. 2 ⊢ {𝐴, 𝐵, 𝐶} = ({𝐴} ∪ {𝐵, 𝐶})
2		wununi.1	. . 3 ⊢ (𝜑 → 𝑈 ∈ WUni)
3		dfsn2 4574	. . . 4 ⊢ {𝐴} = {𝐴, 𝐴}
4		wununi.2	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ 𝑈)
5	2, 4, 4	wunpr 10125	. . . 4 ⊢ (𝜑 → {𝐴, 𝐴} ∈ 𝑈)
6	3, 5	eqeltrid 2917	. . 3 ⊢ (𝜑 → {𝐴} ∈ 𝑈)
7		wunpr.3	. . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝑈)
8		wuntp.3	. . . 4 ⊢ (𝜑 → 𝐶 ∈ 𝑈)
9	2, 7, 8	wunpr 10125	. . 3 ⊢ (𝜑 → {𝐵, 𝐶} ∈ 𝑈)
10	2, 6, 9	wunun 10126	. 2 ⊢ (𝜑 → ({𝐴} ∪ {𝐵, 𝐶}) ∈ 𝑈)
11	1, 10	eqeltrid 2917	1 ⊢ (𝜑 → {𝐴, 𝐵, 𝐶} ∈ 𝑈)

Syntax hints:   → wi 4   ∈ wcel 2110   ∪ cun 3934  {csn 4561  {cpr 4563  {ctp 4565  WUnicwun 10116

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 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2156  ax-12 2172  ax-ext 2793

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-v 3497  df-un 3941  df-in 3943  df-ss 3952  df-sn 4562  df-pr 4564  df-tp 4566  df-uni 4833  df-tr 5166  df-wun 10118

This theorem is referenced by:  catcfuccl  17363  catcxpccl  17451