Metamath Proof Explorer


Theorem mnupwd

Description: Minimal universes are closed under powersets. (Contributed by Rohan Ridenour, 13-Aug-2023)

Ref Expression
Hypotheses mnupwd.1 𝑀 = { 𝑘 ∣ ∀ 𝑙𝑘 ( 𝒫 𝑙𝑘 ∧ ∀ 𝑚𝑛𝑘 ( 𝒫 𝑙𝑛 ∧ ∀ 𝑝𝑙 ( ∃ 𝑞𝑘 ( 𝑝𝑞𝑞𝑚 ) → ∃ 𝑟𝑚 ( 𝑝𝑟 𝑟𝑛 ) ) ) ) }
mnupwd.2 ( 𝜑𝑈𝑀 )
mnupwd.3 ( 𝜑𝐴𝑈 )
Assertion mnupwd ( 𝜑 → 𝒫 𝐴𝑈 )

Proof

Step Hyp Ref Expression
1 mnupwd.1 𝑀 = { 𝑘 ∣ ∀ 𝑙𝑘 ( 𝒫 𝑙𝑘 ∧ ∀ 𝑚𝑛𝑘 ( 𝒫 𝑙𝑛 ∧ ∀ 𝑝𝑙 ( ∃ 𝑞𝑘 ( 𝑝𝑞𝑞𝑚 ) → ∃ 𝑟𝑚 ( 𝑝𝑟 𝑟𝑛 ) ) ) ) }
2 mnupwd.2 ( 𝜑𝑈𝑀 )
3 mnupwd.3 ( 𝜑𝐴𝑈 )
4 0ex ∅ ∈ V
5 4 a1i ( 𝜑 → ∅ ∈ V )
6 1 2 3 5 mnuop23d ( 𝜑 → ∃ 𝑤𝑈 ( 𝒫 𝐴𝑤 ∧ ∀ 𝑖𝐴 ( ∃ 𝑣𝑈 ( 𝑖𝑣𝑣 ∈ ∅ ) → ∃ 𝑢 ∈ ∅ ( 𝑖𝑢 𝑢𝑤 ) ) ) )
7 simpl ( ( 𝒫 𝐴𝑤 ∧ ∀ 𝑖𝐴 ( ∃ 𝑣𝑈 ( 𝑖𝑣𝑣 ∈ ∅ ) → ∃ 𝑢 ∈ ∅ ( 𝑖𝑢 𝑢𝑤 ) ) ) → 𝒫 𝐴𝑤 )
8 7 reximi ( ∃ 𝑤𝑈 ( 𝒫 𝐴𝑤 ∧ ∀ 𝑖𝐴 ( ∃ 𝑣𝑈 ( 𝑖𝑣𝑣 ∈ ∅ ) → ∃ 𝑢 ∈ ∅ ( 𝑖𝑢 𝑢𝑤 ) ) ) → ∃ 𝑤𝑈 𝒫 𝐴𝑤 )
9 6 8 syl ( 𝜑 → ∃ 𝑤𝑈 𝒫 𝐴𝑤 )
10 1 2 9 mnuss2d ( 𝜑 → 𝒫 𝐴𝑈 )