This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

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 )