Theorem gchacg

Description: A "local" form of gchac . If A and ~P A are GCH-sets, then ~P A is well-orderable. The proof is due to Specker. Theorem 2.1 of KanamoriPincus p. 419. (Contributed by Mario Carneiro, 15-May-2015)

	Ref	Expression
Assertion	gchacg	⊢ ( ( ω ≼ 𝐴 ∧ 𝐴 ∈ GCH ∧ 𝒫 𝐴 ∈ GCH ) → 𝒫 𝐴 ∈ dom card )

Proof

Step	Hyp	Ref	Expression
1		harcl	⊢ ( har ‘ 𝐴 ) ∈ On
2		gchhar	⊢ ( ( ω ≼ 𝐴 ∧ 𝐴 ∈ GCH ∧ 𝒫 𝐴 ∈ GCH ) → ( har ‘ 𝐴 ) ≈ 𝒫 𝐴 )
3		isnumi	⊢ ( ( ( har ‘ 𝐴 ) ∈ On ∧ ( har ‘ 𝐴 ) ≈ 𝒫 𝐴 ) → 𝒫 𝐴 ∈ dom card )
4	1 2 3	sylancr	⊢ ( ( ω ≼ 𝐴 ∧ 𝐴 ∈ GCH ∧ 𝒫 𝐴 ∈ GCH ) → 𝒫 𝐴 ∈ dom card )