Theorem elgch 9444

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 9443	. . . 4 ⊢ GCH = (Fin ∪ {𝑦 ∣ ∀𝑥 ¬ (𝑦 ≺ 𝑥 ∧ 𝑥 ≺ 𝒫 𝑦)})
2	1	eleq2i 2693	. . 3 ⊢ (𝐴 ∈ GCH ↔ 𝐴 ∈ (Fin ∪ {𝑦 ∣ ∀𝑥 ¬ (𝑦 ≺ 𝑥 ∧ 𝑥 ≺ 𝒫 𝑦)}))
3		elun 3753	. . 3 ⊢ (𝐴 ∈ (Fin ∪ {𝑦 ∣ ∀𝑥 ¬ (𝑦 ≺ 𝑥 ∧ 𝑥 ≺ 𝒫 𝑦)}) ↔ (𝐴 ∈ Fin ∨ 𝐴 ∈ {𝑦 ∣ ∀𝑥 ¬ (𝑦 ≺ 𝑥 ∧ 𝑥 ≺ 𝒫 𝑦)}))
4	2, 3	bitri 264	. 2 ⊢ (𝐴 ∈ GCH ↔ (𝐴 ∈ Fin ∨ 𝐴 ∈ {𝑦 ∣ ∀𝑥 ¬ (𝑦 ≺ 𝑥 ∧ 𝑥 ≺ 𝒫 𝑦)}))
5		breq1 4656	. . . . . . 7 ⊢ (𝑦 = 𝐴 → (𝑦 ≺ 𝑥 ↔ 𝐴 ≺ 𝑥))
6		pweq 4161	. . . . . . . 8 ⊢ (𝑦 = 𝐴 → 𝒫 𝑦 = 𝒫 𝐴)
7	6	breq2d 4665	. . . . . . 7 ⊢ (𝑦 = 𝐴 → (𝑥 ≺ 𝒫 𝑦 ↔ 𝑥 ≺ 𝒫 𝐴))
8	5, 7	anbi12d 747	. . . . . 6 ⊢ (𝑦 = 𝐴 → ((𝑦 ≺ 𝑥 ∧ 𝑥 ≺ 𝒫 𝑦) ↔ (𝐴 ≺ 𝑥 ∧ 𝑥 ≺ 𝒫 𝐴)))
9	8	notbid 308	. . . . 5 ⊢ (𝑦 = 𝐴 → (¬ (𝑦 ≺ 𝑥 ∧ 𝑥 ≺ 𝒫 𝑦) ↔ ¬ (𝐴 ≺ 𝑥 ∧ 𝑥 ≺ 𝒫 𝐴)))
10	9	albidv 1849	. . . 4 ⊢ (𝑦 = 𝐴 → (∀𝑥 ¬ (𝑦 ≺ 𝑥 ∧ 𝑥 ≺ 𝒫 𝑦) ↔ ∀𝑥 ¬ (𝐴 ≺ 𝑥 ∧ 𝑥 ≺ 𝒫 𝐴)))
11	10	elabg 3351	. . 3 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ {𝑦 ∣ ∀𝑥 ¬ (𝑦 ≺ 𝑥 ∧ 𝑥 ≺ 𝒫 𝑦)} ↔ ∀𝑥 ¬ (𝐴 ≺ 𝑥 ∧ 𝑥 ≺ 𝒫 𝐴)))
12	11	orbi2d 738	. 2 ⊢ (𝐴 ∈ 𝑉 → ((𝐴 ∈ Fin ∨ 𝐴 ∈ {𝑦 ∣ ∀𝑥 ¬ (𝑦 ≺ 𝑥 ∧ 𝑥 ≺ 𝒫 𝑦)}) ↔ (𝐴 ∈ Fin ∨ ∀𝑥 ¬ (𝐴 ≺ 𝑥 ∧ 𝑥 ≺ 𝒫 𝐴))))
13	4, 12	syl5bb 272	1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ GCH ↔ (𝐴 ∈ Fin ∨ ∀𝑥 ¬ (𝐴 ≺ 𝑥 ∧ 𝑥 ≺ 𝒫 𝐴))))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∨ wo 383   ∧ wa 384  ∀wal 1481   = wceq 1483   ∈ wcel 1990  {cab 2608   ∪ cun 3572  𝒫 cpw 4158   class class class wbr 4653   ≺ csdm 7954  Fincfn 7955  GCHcgch 9442

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-gch 9443

This theorem is referenced by:  gchi  9446  engch  9450  hargch  9495  alephgch  9496