Theorem elgch 10543

Description: Elementhood in the collection of GCH-sets. (Contributed by Mario Carneiro, 15-May-2015.)

Assertion

Ref	Expression
elgch	⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ GCH ↔ (𝐴 ∈ Fin ∨ ∀𝑥 ¬ (𝐴 ≺ 𝑥 ∧ 𝑥 ≺ 𝒫 𝐴))))

Distinct variable group:   𝑥,𝐴

Allowed substitution hint:   𝑉(𝑥)

Proof of Theorem elgch

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-gch 10542	. . . 4 ⊢ GCH = (Fin ∪ {𝑦 ∣ ∀𝑥 ¬ (𝑦 ≺ 𝑥 ∧ 𝑥 ≺ 𝒫 𝑦)})
2	1	eleq2i 2832	. . 3 ⊢ (𝐴 ∈ GCH ↔ 𝐴 ∈ (Fin ∪ {𝑦 ∣ ∀𝑥 ¬ (𝑦 ≺ 𝑥 ∧ 𝑥 ≺ 𝒫 𝑦)}))
3		elun 4090	. . 3 ⊢ (𝐴 ∈ (Fin ∪ {𝑦 ∣ ∀𝑥 ¬ (𝑦 ≺ 𝑥 ∧ 𝑥 ≺ 𝒫 𝑦)}) ↔ (𝐴 ∈ Fin ∨ 𝐴 ∈ {𝑦 ∣ ∀𝑥 ¬ (𝑦 ≺ 𝑥 ∧ 𝑥 ≺ 𝒫 𝑦)}))
4	2, 3	bitri 276	. 2 ⊢ (𝐴 ∈ GCH ↔ (𝐴 ∈ Fin ∨ 𝐴 ∈ {𝑦 ∣ ∀𝑥 ¬ (𝑦 ≺ 𝑥 ∧ 𝑥 ≺ 𝒫 𝑦)}))
5		breq1 5082	. . . . . . 7 ⊢ (𝑦 = 𝐴 → (𝑦 ≺ 𝑥 ↔ 𝐴 ≺ 𝑥))
6		pweq 4550	. . . . . . . 8 ⊢ (𝑦 = 𝐴 → 𝒫 𝑦 = 𝒫 𝐴)
7	6	breq2d 5091	. . . . . . 7 ⊢ (𝑦 = 𝐴 → (𝑥 ≺ 𝒫 𝑦 ↔ 𝑥 ≺ 𝒫 𝐴))
8	5, 7	anbi12d 638	. . . . . 6 ⊢ (𝑦 = 𝐴 → ((𝑦 ≺ 𝑥 ∧ 𝑥 ≺ 𝒫 𝑦) ↔ (𝐴 ≺ 𝑥 ∧ 𝑥 ≺ 𝒫 𝐴)))
9	8	notbid 319	. . . . 5 ⊢ (𝑦 = 𝐴 → (¬ (𝑦 ≺ 𝑥 ∧ 𝑥 ≺ 𝒫 𝑦) ↔ ¬ (𝐴 ≺ 𝑥 ∧ 𝑥 ≺ 𝒫 𝐴)))
10	9	albidv 1927	. . . 4 ⊢ (𝑦 = 𝐴 → (∀𝑥 ¬ (𝑦 ≺ 𝑥 ∧ 𝑥 ≺ 𝒫 𝑦) ↔ ∀𝑥 ¬ (𝐴 ≺ 𝑥 ∧ 𝑥 ≺ 𝒫 𝐴)))
11	10	elabg 3621	. . 3 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ {𝑦 ∣ ∀𝑥 ¬ (𝑦 ≺ 𝑥 ∧ 𝑥 ≺ 𝒫 𝑦)} ↔ ∀𝑥 ¬ (𝐴 ≺ 𝑥 ∧ 𝑥 ≺ 𝒫 𝐴)))
12	11	orbi2d 921	. 2 ⊢ (𝐴 ∈ 𝑉 → ((𝐴 ∈ Fin ∨ 𝐴 ∈ {𝑦 ∣ ∀𝑥 ¬ (𝑦 ≺ 𝑥 ∧ 𝑥 ≺ 𝒫 𝑦)}) ↔ (𝐴 ∈ Fin ∨ ∀𝑥 ¬ (𝐴 ≺ 𝑥 ∧ 𝑥 ≺ 𝒫 𝐴))))
13	4, 12	bitrid 284	1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ GCH ↔ (𝐴 ∈ Fin ∨ ∀𝑥 ¬ (𝐴 ≺ 𝑥 ∧ 𝑥 ≺ 𝒫 𝐴))))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 207   ∧ wa 396   ∨ wo 853  ∀wal 1545   = wceq 1547   ∈ wcel 2119  {cab 2718   ∪ cun 3888  𝒫 cpw 4536   class class class wbr 5079   ≺ csdm 8889  Fincfn 8890  GCHcgch 10541

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 2712

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 2719  df-cleq 2732  df-clel 2815  df-rab 3393  df-v 3434  df-dif 3893  df-un 3895  df-ss 3907  df-nul 4269  df-if 4462  df-pw 4538  df-sn 4563  df-pr 4565  df-op 4569  df-br 5080  df-gch 10542

This theorem is referenced by:  gchi  10545  engch  10549  hargch  10594  alephgch  10595