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Metamath Proof Explorer

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 )