Theorem wuntp 9819

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 4474	. 2 ⊢ {𝐴, 𝐵, 𝐶} = ({𝐴} ∪ {𝐵, 𝐶})
2		wununi.1	. . 3 ⊢ (𝜑 → 𝑈 ∈ WUni)
3		dfsn2 4379	. . . 4 ⊢ {𝐴} = {𝐴, 𝐴}
4		wununi.2	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ 𝑈)
5	2, 4, 4	wunpr 9817	. . . 4 ⊢ (𝜑 → {𝐴, 𝐴} ∈ 𝑈)
6	3, 5	syl5eqel 2880	. . 3 ⊢ (𝜑 → {𝐴} ∈ 𝑈)
7		wunpr.3	. . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝑈)
8		wuntp.3	. . . 4 ⊢ (𝜑 → 𝐶 ∈ 𝑈)
9	2, 7, 8	wunpr 9817	. . 3 ⊢ (𝜑 → {𝐵, 𝐶} ∈ 𝑈)
10	2, 6, 9	wunun 9818	. 2 ⊢ (𝜑 → ({𝐴} ∪ {𝐵, 𝐶}) ∈ 𝑈)
11	1, 10	syl5eqel 2880	1 ⊢ (𝜑 → {𝐴, 𝐵, 𝐶} ∈ 𝑈)

Syntax hints:   → wi 4   ∈ wcel 2157   ∪ cun 3765  {csn 4366  {cpr 4368  {ctp 4370  WUnicwun 9808

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2354  ax-ext 2775

This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3or 1109  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-clab 2784  df-cleq 2790  df-clel 2793  df-nfc 2928  df-ne 2970  df-ral 3092  df-rex 3093  df-v 3385  df-un 3772  df-in 3774  df-ss 3781  df-sn 4367  df-pr 4369  df-tp 4371  df-uni 4627  df-tr 4944  df-wun 9810

This theorem is referenced by:  catcfuccl  17070  catcxpccl  17159