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Metamath Proof Explorer

Theorem wunss

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

Ref Expression
Hypotheses wununi.1 ⊢ ( 𝜑 → 𝑈 ∈ WUni )
wununi.2 ⊢ ( 𝜑 → 𝐴 ∈ 𝑈 )
wunss.3 ⊢ ( 𝜑 → 𝐵 ⊆ 𝐴 )
Assertion wunss ( 𝜑 → 𝐵 ∈ 𝑈 )

Proof

Step Hyp Ref Expression
1 wununi.1 ⊢ ( 𝜑 → 𝑈 ∈ WUni )
2 wununi.2 ⊢ ( 𝜑 → 𝐴 ∈ 𝑈 )
3 wunss.3 ⊢ ( 𝜑 → 𝐵 ⊆ 𝐴 )
4 1 2 wunpw ⊢ ( 𝜑 → 𝒫 𝐴 ∈ 𝑈 )
5 1 4 wunelss ⊢ ( 𝜑 → 𝒫 𝐴 ⊆ 𝑈 )
6 2 3 sselpwd ⊢ ( 𝜑 → 𝐵 ∈ 𝒫 𝐴 )
7 5 6 sseldd ⊢ ( 𝜑 → 𝐵 ∈ 𝑈 )