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Metamath Proof Explorer

Theorem gchi

Description: The only GCH-sets which have other sets between it and its power set are finite sets. (Contributed by Mario Carneiro, 15-May-2015)

Ref Expression
Assertion gchi ( ( 𝐴 ∈ GCH ∧ 𝐴 ≺ 𝐵 ∧ 𝐵 ≺ 𝒫 𝐴 ) → 𝐴 ∈ Fin )

Proof

Step Hyp Ref Expression
1 relsdom ⊢ Rel ≺
2 1 brrelex1i ⊢ ( 𝐵 ≺ 𝒫 𝐴 → 𝐵 ∈ V )
3 2 adantl ⊢ ( ( 𝐴 ≺ 𝐵 ∧ 𝐵 ≺ 𝒫 𝐴 ) → 𝐵 ∈ V )
4 breq2 ⊢ ( 𝑥 = 𝐵 → ( 𝐴 ≺ 𝑥 ↔ 𝐴 ≺ 𝐵 ) )
5 breq1 ⊢ ( 𝑥 = 𝐵 → ( 𝑥 ≺ 𝒫 𝐴 ↔ 𝐵 ≺ 𝒫 𝐴 ) )
6 4 5 anbi12d ⊢ ( 𝑥 = 𝐵 → ( ( 𝐴 ≺ 𝑥 ∧ 𝑥 ≺ 𝒫 𝐴 ) ↔ ( 𝐴 ≺ 𝐵 ∧ 𝐵 ≺ 𝒫 𝐴 ) ) )
7 6 spcegv ⊢ ( 𝐵 ∈ V → ( ( 𝐴 ≺ 𝐵 ∧ 𝐵 ≺ 𝒫 𝐴 ) → ∃ 𝑥 ( 𝐴 ≺ 𝑥 ∧ 𝑥 ≺ 𝒫 𝐴 ) ) )
8 3 7 mpcom ⊢ ( ( 𝐴 ≺ 𝐵 ∧ 𝐵 ≺ 𝒫 𝐴 ) → ∃ 𝑥 ( 𝐴 ≺ 𝑥 ∧ 𝑥 ≺ 𝒫 𝐴 ) )
9 df-ex ⊢ ( ∃ 𝑥 ( 𝐴 ≺ 𝑥 ∧ 𝑥 ≺ 𝒫 𝐴 ) ↔ ¬ ∀ 𝑥 ¬ ( 𝐴 ≺ 𝑥 ∧ 𝑥 ≺ 𝒫 𝐴 ) )
10 8 9 sylib ⊢ ( ( 𝐴 ≺ 𝐵 ∧ 𝐵 ≺ 𝒫 𝐴 ) → ¬ ∀ 𝑥 ¬ ( 𝐴 ≺ 𝑥 ∧ 𝑥 ≺ 𝒫 𝐴 ) )
11 elgch ⊢ ( 𝐴 ∈ GCH → ( 𝐴 ∈ GCH ↔ ( 𝐴 ∈ Fin ∨ ∀ 𝑥 ¬ ( 𝐴 ≺ 𝑥 ∧ 𝑥 ≺ 𝒫 𝐴 ) ) ) )
12 11 ibi ⊢ ( 𝐴 ∈ GCH → ( 𝐴 ∈ Fin ∨ ∀ 𝑥 ¬ ( 𝐴 ≺ 𝑥 ∧ 𝑥 ≺ 𝒫 𝐴 ) ) )
13 12 orcomd ⊢ ( 𝐴 ∈ GCH → ( ∀ 𝑥 ¬ ( 𝐴 ≺ 𝑥 ∧ 𝑥 ≺ 𝒫 𝐴 ) ∨ 𝐴 ∈ Fin ) )
14 13 ord ⊢ ( 𝐴 ∈ GCH → ( ¬ ∀ 𝑥 ¬ ( 𝐴 ≺ 𝑥 ∧ 𝑥 ≺ 𝒫 𝐴 ) → 𝐴 ∈ Fin ) )
15 10 14 syl5 ⊢ ( 𝐴 ∈ GCH → ( ( 𝐴 ≺ 𝐵 ∧ 𝐵 ≺ 𝒫 𝐴 ) → 𝐴 ∈ Fin ) )
16 15 3impib ⊢ ( ( 𝐴 ∈ GCH ∧ 𝐴 ≺ 𝐵 ∧ 𝐵 ≺ 𝒫 𝐴 ) → 𝐴 ∈ Fin )