Theorem gchi 10693

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.)

Assertion

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

Proof of Theorem gchi

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		relsdom 9010	. . . . . . 7 ⊢ Rel ≺
2	1	brrelex1i 5756	. . . . . 6 ⊢ (𝐵 ≺ 𝒫 𝐴 → 𝐵 ∈ V)
3	2	adantl 481	. . . . 5 ⊢ ((𝐴 ≺ 𝐵 ∧ 𝐵 ≺ 𝒫 𝐴) → 𝐵 ∈ V)
4		breq2 5170	. . . . . . 7 ⊢ (𝑥 = 𝐵 → (𝐴 ≺ 𝑥 ↔ 𝐴 ≺ 𝐵))
5		breq1 5169	. . . . . . 7 ⊢ (𝑥 = 𝐵 → (𝑥 ≺ 𝒫 𝐴 ↔ 𝐵 ≺ 𝒫 𝐴))
6	4, 5	anbi12d 631	. . . . . 6 ⊢ (𝑥 = 𝐵 → ((𝐴 ≺ 𝑥 ∧ 𝑥 ≺ 𝒫 𝐴) ↔ (𝐴 ≺ 𝐵 ∧ 𝐵 ≺ 𝒫 𝐴)))
7	6	spcegv 3610	. . . . 5 ⊢ (𝐵 ∈ V → ((𝐴 ≺ 𝐵 ∧ 𝐵 ≺ 𝒫 𝐴) → ∃𝑥(𝐴 ≺ 𝑥 ∧ 𝑥 ≺ 𝒫 𝐴)))
8	3, 7	mpcom 38	. . . 4 ⊢ ((𝐴 ≺ 𝐵 ∧ 𝐵 ≺ 𝒫 𝐴) → ∃𝑥(𝐴 ≺ 𝑥 ∧ 𝑥 ≺ 𝒫 𝐴))
9		df-ex 1778	. . . 4 ⊢ (∃𝑥(𝐴 ≺ 𝑥 ∧ 𝑥 ≺ 𝒫 𝐴) ↔ ¬ ∀𝑥 ¬ (𝐴 ≺ 𝑥 ∧ 𝑥 ≺ 𝒫 𝐴))
10	8, 9	sylib 218	. . 3 ⊢ ((𝐴 ≺ 𝐵 ∧ 𝐵 ≺ 𝒫 𝐴) → ¬ ∀𝑥 ¬ (𝐴 ≺ 𝑥 ∧ 𝑥 ≺ 𝒫 𝐴))
11		elgch 10691	. . . . . 6 ⊢ (𝐴 ∈ GCH → (𝐴 ∈ GCH ↔ (𝐴 ∈ Fin ∨ ∀𝑥 ¬ (𝐴 ≺ 𝑥 ∧ 𝑥 ≺ 𝒫 𝐴))))
12	11	ibi 267	. . . . 5 ⊢ (𝐴 ∈ GCH → (𝐴 ∈ Fin ∨ ∀𝑥 ¬ (𝐴 ≺ 𝑥 ∧ 𝑥 ≺ 𝒫 𝐴)))
13	12	orcomd 870	. . . 4 ⊢ (𝐴 ∈ GCH → (∀𝑥 ¬ (𝐴 ≺ 𝑥 ∧ 𝑥 ≺ 𝒫 𝐴) ∨ 𝐴 ∈ Fin))
14	13	ord 863	. . 3 ⊢ (𝐴 ∈ GCH → (¬ ∀𝑥 ¬ (𝐴 ≺ 𝑥 ∧ 𝑥 ≺ 𝒫 𝐴) → 𝐴 ∈ Fin))
15	10, 14	syl5 34	. 2 ⊢ (𝐴 ∈ GCH → ((𝐴 ≺ 𝐵 ∧ 𝐵 ≺ 𝒫 𝐴) → 𝐴 ∈ Fin))
16	15	3impib 1116	1 ⊢ ((𝐴 ∈ GCH ∧ 𝐴 ≺ 𝐵 ∧ 𝐵 ≺ 𝒫 𝐴) → 𝐴 ∈ Fin)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   ∨ wo 846   ∧ w3a 1087  ∀wal 1535   = wceq 1537  ∃wex 1777   ∈ wcel 2108  Vcvv 3488  𝒫 cpw 4622   class class class wbr 5166   ≺ csdm 9002  Fincfn 9003  GCHcgch 10689

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-br 5167  df-opab 5229  df-xp 5706  df-rel 5707  df-dom 9005  df-sdom 9006  df-gch 10690

This theorem is referenced by:  gchen1  10694  gchen2  10695  gchpwdom  10739  gchaleph  10740