Theorem pwinfi2

Description: The powerclass of an infinite set is an infinite set, and vice-versa. Here U is a weak universe. (Contributed by RP, 21-Mar-2020)

	Ref	Expression
Assertion	pwinfi2	⊢ ( 𝑈 ∈ WUni → ( 𝐴 ∈ ( 𝑈 ∖ Fin ) ↔ 𝒫 𝐴 ∈ ( 𝑈 ∖ Fin ) ) )

Proof

Step	Hyp	Ref	Expression
1		iswun	⊢ ( 𝑈 ∈ WUni → ( 𝑈 ∈ WUni ↔ ( Tr 𝑈 ∧ 𝑈 ≠ ∅ ∧ ∀ 𝑥 ∈ 𝑈 ( ∪ 𝑥 ∈ 𝑈 ∧ 𝒫 𝑥 ∈ 𝑈 ∧ ∀ 𝑦 ∈ 𝑈 { 𝑥 , 𝑦 } ∈ 𝑈 ) ) ) )
2	1	ibi	⊢ ( 𝑈 ∈ WUni → ( Tr 𝑈 ∧ 𝑈 ≠ ∅ ∧ ∀ 𝑥 ∈ 𝑈 ( ∪ 𝑥 ∈ 𝑈 ∧ 𝒫 𝑥 ∈ 𝑈 ∧ ∀ 𝑦 ∈ 𝑈 { 𝑥 , 𝑦 } ∈ 𝑈 ) ) )
3		3simpa	⊢ ( ( ∪ 𝑥 ∈ 𝑈 ∧ 𝒫 𝑥 ∈ 𝑈 ∧ ∀ 𝑦 ∈ 𝑈 { 𝑥 , 𝑦 } ∈ 𝑈 ) → ( ∪ 𝑥 ∈ 𝑈 ∧ 𝒫 𝑥 ∈ 𝑈 ) )
4	3	ralimi	⊢ ( ∀ 𝑥 ∈ 𝑈 ( ∪ 𝑥 ∈ 𝑈 ∧ 𝒫 𝑥 ∈ 𝑈 ∧ ∀ 𝑦 ∈ 𝑈 { 𝑥 , 𝑦 } ∈ 𝑈 ) → ∀ 𝑥 ∈ 𝑈 ( ∪ 𝑥 ∈ 𝑈 ∧ 𝒫 𝑥 ∈ 𝑈 ) )
5	4	3ad2ant3	⊢ ( ( Tr 𝑈 ∧ 𝑈 ≠ ∅ ∧ ∀ 𝑥 ∈ 𝑈 ( ∪ 𝑥 ∈ 𝑈 ∧ 𝒫 𝑥 ∈ 𝑈 ∧ ∀ 𝑦 ∈ 𝑈 { 𝑥 , 𝑦 } ∈ 𝑈 ) ) → ∀ 𝑥 ∈ 𝑈 ( ∪ 𝑥 ∈ 𝑈 ∧ 𝒫 𝑥 ∈ 𝑈 ) )
6		pwinfig	⊢ ( ∀ 𝑥 ∈ 𝑈 ( ∪ 𝑥 ∈ 𝑈 ∧ 𝒫 𝑥 ∈ 𝑈 ) → ( 𝐴 ∈ ( 𝑈 ∖ Fin ) ↔ 𝒫 𝐴 ∈ ( 𝑈 ∖ Fin ) ) )
7	2 5 6	3syl	⊢ ( 𝑈 ∈ WUni → ( 𝐴 ∈ ( 𝑈 ∖ Fin ) ↔ 𝒫 𝐴 ∈ ( 𝑈 ∖ Fin ) ) )