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Metamath Proof Explorer

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