Theorem gchen1

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

	Ref	Expression
Assertion	gchen1	⊢ ( ( ( 𝐴 ∈ GCH ∧ ¬ 𝐴 ∈ Fin ) ∧ ( 𝐴 ≼ 𝐵 ∧ 𝐵 ≺ 𝒫 𝐴 ) ) → 𝐴 ≈ 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		simprl	⊢ ( ( ( 𝐴 ∈ GCH ∧ ¬ 𝐴 ∈ Fin ) ∧ ( 𝐴 ≼ 𝐵 ∧ 𝐵 ≺ 𝒫 𝐴 ) ) → 𝐴 ≼ 𝐵 )
2		gchi	⊢ ( ( 𝐴 ∈ GCH ∧ 𝐴 ≺ 𝐵 ∧ 𝐵 ≺ 𝒫 𝐴 ) → 𝐴 ∈ Fin )
3	2	3com23	⊢ ( ( 𝐴 ∈ GCH ∧ 𝐵 ≺ 𝒫 𝐴 ∧ 𝐴 ≺ 𝐵 ) → 𝐴 ∈ Fin )
4	3	3expia	⊢ ( ( 𝐴 ∈ GCH ∧ 𝐵 ≺ 𝒫 𝐴 ) → ( 𝐴 ≺ 𝐵 → 𝐴 ∈ Fin ) )
5	4	con3dimp	⊢ ( ( ( 𝐴 ∈ GCH ∧ 𝐵 ≺ 𝒫 𝐴 ) ∧ ¬ 𝐴 ∈ Fin ) → ¬ 𝐴 ≺ 𝐵 )
6	5	an32s	⊢ ( ( ( 𝐴 ∈ GCH ∧ ¬ 𝐴 ∈ Fin ) ∧ 𝐵 ≺ 𝒫 𝐴 ) → ¬ 𝐴 ≺ 𝐵 )
7	6	adantrl	⊢ ( ( ( 𝐴 ∈ GCH ∧ ¬ 𝐴 ∈ Fin ) ∧ ( 𝐴 ≼ 𝐵 ∧ 𝐵 ≺ 𝒫 𝐴 ) ) → ¬ 𝐴 ≺ 𝐵 )
8		bren2	⊢ ( 𝐴 ≈ 𝐵 ↔ ( 𝐴 ≼ 𝐵 ∧ ¬ 𝐴 ≺ 𝐵 ) )
9	1 7 8	sylanbrc	⊢ ( ( ( 𝐴 ∈ GCH ∧ ¬ 𝐴 ∈ Fin ) ∧ ( 𝐴 ≼ 𝐵 ∧ 𝐵 ≺ 𝒫 𝐴 ) ) → 𝐴 ≈ 𝐵 )