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Metamath Proof Explorer

Theorem wunsn

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

Ref Expression
Hypotheses wununi.1 ⊢ ( 𝜑 → 𝑈 ∈ WUni )
wununi.2 ⊢ ( 𝜑 → 𝐴 ∈ 𝑈 )
Assertion wunsn ( 𝜑 → { 𝐴 } ∈ 𝑈 )

Proof

Step Hyp Ref Expression
1 wununi.1 ⊢ ( 𝜑 → 𝑈 ∈ WUni )
2 wununi.2 ⊢ ( 𝜑 → 𝐴 ∈ 𝑈 )
3 dfsn2 ⊢ { 𝐴 } = { 𝐴 , 𝐴 }
4 1 2 2 wunpr ⊢ ( 𝜑 → { 𝐴 , 𝐴 } ∈ 𝑈 )
5 3 4 eqeltrid ⊢ ( 𝜑 → { 𝐴 } ∈ 𝑈 )