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Metamath Proof Explorer

Theorem gchor

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

Ref Expression
Assertion gchor ( ( ( 𝐴 ∈ GCH ∧ ¬ 𝐴 ∈ Fin ) ∧ ( 𝐴 ≼ 𝐵 ∧ 𝐵 ≼ 𝒫 𝐴 ) ) → ( 𝐴 ≈ 𝐵 ∨ 𝐵 ≈ 𝒫 𝐴 ) )

Proof

Step Hyp Ref Expression
1 simprr ⊢ ( ( ( 𝐴 ∈ GCH ∧ ¬ 𝐴 ∈ Fin ) ∧ ( 𝐴 ≼ 𝐵 ∧ 𝐵 ≼ 𝒫 𝐴 ) ) → 𝐵 ≼ 𝒫 𝐴 )
2 brdom2 ⊢ ( 𝐵 ≼ 𝒫 𝐴 ↔ ( 𝐵 ≺ 𝒫 𝐴 ∨ 𝐵 ≈ 𝒫 𝐴 ) )
3 1 2 sylib ⊢ ( ( ( 𝐴 ∈ GCH ∧ ¬ 𝐴 ∈ Fin ) ∧ ( 𝐴 ≼ 𝐵 ∧ 𝐵 ≼ 𝒫 𝐴 ) ) → ( 𝐵 ≺ 𝒫 𝐴 ∨ 𝐵 ≈ 𝒫 𝐴 ) )
4 gchen1 ⊢ ( ( ( 𝐴 ∈ GCH ∧ ¬ 𝐴 ∈ Fin ) ∧ ( 𝐴 ≼ 𝐵 ∧ 𝐵 ≺ 𝒫 𝐴 ) ) → 𝐴 ≈ 𝐵 )
5 4 expr ⊢ ( ( ( 𝐴 ∈ GCH ∧ ¬ 𝐴 ∈ Fin ) ∧ 𝐴 ≼ 𝐵 ) → ( 𝐵 ≺ 𝒫 𝐴 → 𝐴 ≈ 𝐵 ) )
6 5 adantrr ⊢ ( ( ( 𝐴 ∈ GCH ∧ ¬ 𝐴 ∈ Fin ) ∧ ( 𝐴 ≼ 𝐵 ∧ 𝐵 ≼ 𝒫 𝐴 ) ) → ( 𝐵 ≺ 𝒫 𝐴 → 𝐴 ≈ 𝐵 ) )
7 6 orim1d ⊢ ( ( ( 𝐴 ∈ GCH ∧ ¬ 𝐴 ∈ Fin ) ∧ ( 𝐴 ≼ 𝐵 ∧ 𝐵 ≼ 𝒫 𝐴 ) ) → ( ( 𝐵 ≺ 𝒫 𝐴 ∨ 𝐵 ≈ 𝒫 𝐴 ) → ( 𝐴 ≈ 𝐵 ∨ 𝐵 ≈ 𝒫 𝐴 ) ) )
8 3 7 mpd ⊢ ( ( ( 𝐴 ∈ GCH ∧ ¬ 𝐴 ∈ Fin ) ∧ ( 𝐴 ≼ 𝐵 ∧ 𝐵 ≼ 𝒫 𝐴 ) ) → ( 𝐴 ≈ 𝐵 ∨ 𝐵 ≈ 𝒫 𝐴 ) )