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Metamath Proof Explorer

Theorem wunot

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

Ref Expression
Hypotheses wun0.1 ⊢ φ → U ∈ WUni
wunop.2 ⊢ φ → A ∈ U
wunop.3 ⊢ φ → B ∈ U
wunot.3 ⊢ φ → C ∈ U
Assertion wunot ⊢ φ → A B C ∈ U

Proof

Step Hyp Ref Expression
1 wun0.1 ⊢ φ → U ∈ WUni
2 wunop.2 ⊢ φ → A ∈ U
3 wunop.3 ⊢ φ → B ∈ U
4 wunot.3 ⊢ φ → C ∈ U
5 df-ot ⊢ A B C = A B C
6 1 2 3 wunop ⊢ φ → A B ∈ U
7 1 6 4 wunop ⊢ φ → A B C ∈ U
8 5 7 eqeltrid ⊢ φ → A B C ∈ U