Theorem wuntp

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

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

Proof

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