Theorem gchi 10538

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 8890	. . . . . . 7 ⊢ Rel ≺
2	1	brrelex1i 5674	. . . . . 6 ⊢ (𝐵 ≺ 𝒫 𝐴 → 𝐵 ∈ V)
3	2	adantl 482	. . . . 5 ⊢ ((𝐴 ≺ 𝐵 ∧ 𝐵 ≺ 𝒫 𝐴) → 𝐵 ∈ V)
4		breq2 5076	. . . . . . 7 ⊢ (𝑥 = 𝐵 → (𝐴 ≺ 𝑥 ↔ 𝐴 ≺ 𝐵))
5		breq1 5075	. . . . . . 7 ⊢ (𝑥 = 𝐵 → (𝑥 ≺ 𝒫 𝐴 ↔ 𝐵 ≺ 𝒫 𝐴))
6	4, 5	anbi12d 638	. . . . . 6 ⊢ (𝑥 = 𝐵 → ((𝐴 ≺ 𝑥 ∧ 𝑥 ≺ 𝒫 𝐴) ↔ (𝐴 ≺ 𝐵 ∧ 𝐵 ≺ 𝒫 𝐴)))
7	6	spcegv 3535	. . . . 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 1787	. . . 4 ⊢ (∃𝑥(𝐴 ≺ 𝑥 ∧ 𝑥 ≺ 𝒫 𝐴) ↔ ¬ ∀𝑥 ¬ (𝐴 ≺ 𝑥 ∧ 𝑥 ≺ 𝒫 𝐴))
10	8, 9	sylib 219	. . 3 ⊢ ((𝐴 ≺ 𝐵 ∧ 𝐵 ≺ 𝒫 𝐴) → ¬ ∀𝑥 ¬ (𝐴 ≺ 𝑥 ∧ 𝑥 ≺ 𝒫 𝐴))
11		elgch 10536	. . . . . 6 ⊢ (𝐴 ∈ GCH → (𝐴 ∈ GCH ↔ (𝐴 ∈ Fin ∨ ∀𝑥 ¬ (𝐴 ≺ 𝑥 ∧ 𝑥 ≺ 𝒫 𝐴))))
12	11	ibi 268	. . . . 5 ⊢ (𝐴 ∈ GCH → (𝐴 ∈ Fin ∨ ∀𝑥 ¬ (𝐴 ≺ 𝑥 ∧ 𝑥 ≺ 𝒫 𝐴)))
13	12	orcomd 877	. . . 4 ⊢ (𝐴 ∈ GCH → (∀𝑥 ¬ (𝐴 ≺ 𝑥 ∧ 𝑥 ≺ 𝒫 𝐴) ∨ 𝐴 ∈ Fin))
14	13	ord 870	. . 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 1122	1 ⊢ ((𝐴 ∈ GCH ∧ 𝐴 ≺ 𝐵 ∧ 𝐵 ≺ 𝒫 𝐴) → 𝐴 ∈ Fin)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 396   ∨ wo 853   ∧ w3a 1092  ∀wal 1545   = wceq 1547  ∃wex 1786   ∈ wcel 2119  Vcvv 3431  𝒫 cpw 4529   class class class wbr 5072   ≺ csdm 8882  Fincfn 8883  GCHcgch 10534

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2711  ax-sep 5218  ax-pr 5362

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-sb 2074  df-clab 2718  df-cleq 2731  df-clel 2814  df-ral 3054  df-rex 3064  df-rab 3392  df-v 3433  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4262  df-if 4455  df-pw 4531  df-sn 4556  df-pr 4558  df-op 4562  df-br 5073  df-opab 5135  df-xp 5624  df-rel 5625  df-dom 8885  df-sdom 8886  df-gch 10535

This theorem is referenced by:  gchen1  10539  gchen2  10540  gchpwdom  10584  gchaleph  10585