Theorem gchen2

Description: If A < B <_ ~P A , and A is an infinite GCH-set, then B = ~P A in cardinality. (Contributed by Mario Carneiro, 15-May-2015)

	Ref	Expression
Assertion	gchen2	⊢ ( ( ( 𝐴 ∈ GCH ∧ ¬ 𝐴 ∈ Fin ) ∧ ( 𝐴 ≺ 𝐵 ∧ 𝐵 ≼ 𝒫 𝐴 ) ) → 𝐵 ≈ 𝒫 𝐴 )

Proof

Step	Hyp	Ref	Expression
1		simprr	⊢ ( ( ( 𝐴 ∈ GCH ∧ ¬ 𝐴 ∈ Fin ) ∧ ( 𝐴 ≺ 𝐵 ∧ 𝐵 ≼ 𝒫 𝐴 ) ) → 𝐵 ≼ 𝒫 𝐴 )
2		gchi	⊢ ( ( 𝐴 ∈ GCH ∧ 𝐴 ≺ 𝐵 ∧ 𝐵 ≺ 𝒫 𝐴 ) → 𝐴 ∈ Fin )
3	2	3expia	⊢ ( ( 𝐴 ∈ GCH ∧ 𝐴 ≺ 𝐵 ) → ( 𝐵 ≺ 𝒫 𝐴 → 𝐴 ∈ Fin ) )
4	3	con3dimp	⊢ ( ( ( 𝐴 ∈ GCH ∧ 𝐴 ≺ 𝐵 ) ∧ ¬ 𝐴 ∈ Fin ) → ¬ 𝐵 ≺ 𝒫 𝐴 )
5	4	an32s	⊢ ( ( ( 𝐴 ∈ GCH ∧ ¬ 𝐴 ∈ Fin ) ∧ 𝐴 ≺ 𝐵 ) → ¬ 𝐵 ≺ 𝒫 𝐴 )
6	5	adantrr	⊢ ( ( ( 𝐴 ∈ GCH ∧ ¬ 𝐴 ∈ Fin ) ∧ ( 𝐴 ≺ 𝐵 ∧ 𝐵 ≼ 𝒫 𝐴 ) ) → ¬ 𝐵 ≺ 𝒫 𝐴 )
7		bren2	⊢ ( 𝐵 ≈ 𝒫 𝐴 ↔ ( 𝐵 ≼ 𝒫 𝐴 ∧ ¬ 𝐵 ≺ 𝒫 𝐴 ) )
8	1 6 7	sylanbrc	⊢ ( ( ( 𝐴 ∈ GCH ∧ ¬ 𝐴 ∈ Fin ) ∧ ( 𝐴 ≺ 𝐵 ∧ 𝐵 ≼ 𝒫 𝐴 ) ) → 𝐵 ≈ 𝒫 𝐴 )