Theorem wuntp 10625

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 4697	. 2 ⊢ {𝐴, 𝐵, 𝐶} = ({𝐴} ∪ {𝐵, 𝐶})
2		wununi.1	. . 3 ⊢ (𝜑 → 𝑈 ∈ WUni)
3		dfsn2 4581	. . . 4 ⊢ {𝐴} = {𝐴, 𝐴}
4		wununi.2	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ 𝑈)
5	2, 4, 4	wunpr 10623	. . . 4 ⊢ (𝜑 → {𝐴, 𝐴} ∈ 𝑈)
6	3, 5	eqeltrid 2841	. . 3 ⊢ (𝜑 → {𝐴} ∈ 𝑈)
7		wunpr.3	. . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝑈)
8		wuntp.3	. . . 4 ⊢ (𝜑 → 𝐶 ∈ 𝑈)
9	2, 7, 8	wunpr 10623	. . 3 ⊢ (𝜑 → {𝐵, 𝐶} ∈ 𝑈)
10	2, 6, 9	wunun 10624	. 2 ⊢ (𝜑 → ({𝐴} ∪ {𝐵, 𝐶}) ∈ 𝑈)
11	1, 10	eqeltrid 2841	1 ⊢ (𝜑 → {𝐴, 𝐵, 𝐶} ∈ 𝑈)

Syntax hints:   → wi 4   ∈ wcel 2114   ∪ cun 3888  {csn 4568  {cpr 4570  {ctp 4572  WUnicwun 10614

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ne 2934  df-ral 3053  df-rex 3063  df-v 3432  df-un 3895  df-ss 3907  df-sn 4569  df-pr 4571  df-tp 4573  df-uni 4852  df-tr 5194  df-wun 10616

This theorem is referenced by:  catcfuccl  18076  catcxpccl  18164