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 )