Theorem elgch 10711

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 10710	. . . 4 ⊢ GCH = (Fin ∪ {𝑦 ∣ ∀𝑥 ¬ (𝑦 ≺ 𝑥 ∧ 𝑥 ≺ 𝒫 𝑦)})
2	1	eleq2i 2852	. . 3 ⊢ (𝐴 ∈ GCH ↔ 𝐴 ∈ (Fin ∪ {𝑦 ∣ ∀𝑥 ¬ (𝑦 ≺ 𝑥 ∧ 𝑥 ≺ 𝒫 𝑦)}))
3		elun 4152	. . 3 ⊢ (𝐴 ∈ (Fin ∪ {𝑦 ∣ ∀𝑥 ¬ (𝑦 ≺ 𝑥 ∧ 𝑥 ≺ 𝒫 𝑦)}) ↔ (𝐴 ∈ Fin ∨ 𝐴 ∈ {𝑦 ∣ ∀𝑥 ¬ (𝑦 ≺ 𝑥 ∧ 𝑥 ≺ 𝒫 𝑦)}))
4	2, 3	bitri 278	. 2 ⊢ (𝐴 ∈ GCH ↔ (𝐴 ∈ Fin ∨ 𝐴 ∈ {𝑦 ∣ ∀𝑥 ¬ (𝑦 ≺ 𝑥 ∧ 𝑥 ≺ 𝒫 𝑦)}))
5		breq1 5105	. . . . . . 7 ⊢ (𝑦 = 𝐴 → (𝑦 ≺ 𝑥 ↔ 𝐴 ≺ 𝑥))
6		pweq 4570	. . . . . . . 8 ⊢ (𝑦 = 𝐴 → 𝒫 𝑦 = 𝒫 𝐴)
7	6	breq2d 5114	. . . . . . 7 ⊢ (𝑦 = 𝐴 → (𝑥 ≺ 𝒫 𝑦 ↔ 𝑥 ≺ 𝒫 𝐴))
8	5, 7	anbi12d 644	. . . . . 6 ⊢ (𝑦 = 𝐴 → ((𝑦 ≺ 𝑥 ∧ 𝑥 ≺ 𝒫 𝑦) ↔ (𝐴 ≺ 𝑥 ∧ 𝑥 ≺ 𝒫 𝐴)))
9	8	notbid 321	. . . . 5 ⊢ (𝑦 = 𝐴 → (¬ (𝑦 ≺ 𝑥 ∧ 𝑥 ≺ 𝒫 𝑦) ↔ ¬ (𝐴 ≺ 𝑥 ∧ 𝑥 ≺ 𝒫 𝐴)))
10	9	albidv 1952	. . . 4 ⊢ (𝑦 = 𝐴 → (∀𝑥 ¬ (𝑦 ≺ 𝑥 ∧ 𝑥 ≺ 𝒫 𝑦) ↔ ∀𝑥 ¬ (𝐴 ≺ 𝑥 ∧ 𝑥 ≺ 𝒫 𝐴)))
11	10	elabg 3629	. . 3 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ {𝑦 ∣ ∀𝑥 ¬ (𝑦 ≺ 𝑥 ∧ 𝑥 ≺ 𝒫 𝑦)} ↔ ∀𝑥 ¬ (𝐴 ≺ 𝑥 ∧ 𝑥 ≺ 𝒫 𝐴)))
12	11	orbi2d 929	. 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 286	1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ GCH ↔ (𝐴 ∈ Fin ∨ ∀𝑥 ¬ (𝐴 ≺ 𝑥 ∧ 𝑥 ≺ 𝒫 𝐴))))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861  ∀wal 1567   = wceq 1569   ∈ wcel 2144  {cab 2738   ∪ cun 4149  𝒫 cpw 4556   class class class wbr 5102   ≺ csdm 8976  Fincfn 8977  GCHcgch 10709

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-rab 3413  df-v 3452  df-ss 3896  df-dif 4047  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-br 5103  df-gch 10710

This theorem is used by:  gchi  10713  engch  10717  hargch  10762  alephgch  10763