Theorem elgch 10457

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 10456	. . . 4 ⊢ GCH = (Fin ∪ {𝑦 ∣ ∀𝑥 ¬ (𝑦 ≺ 𝑥 ∧ 𝑥 ≺ 𝒫 𝑦)})
2	1	eleq2i 2828	. . 3 ⊢ (𝐴 ∈ GCH ↔ 𝐴 ∈ (Fin ∪ {𝑦 ∣ ∀𝑥 ¬ (𝑦 ≺ 𝑥 ∧ 𝑥 ≺ 𝒫 𝑦)}))
3		elun 4093	. . 3 ⊢ (𝐴 ∈ (Fin ∪ {𝑦 ∣ ∀𝑥 ¬ (𝑦 ≺ 𝑥 ∧ 𝑥 ≺ 𝒫 𝑦)}) ↔ (𝐴 ∈ Fin ∨ 𝐴 ∈ {𝑦 ∣ ∀𝑥 ¬ (𝑦 ≺ 𝑥 ∧ 𝑥 ≺ 𝒫 𝑦)}))
4	2, 3	bitri 274	. 2 ⊢ (𝐴 ∈ GCH ↔ (𝐴 ∈ Fin ∨ 𝐴 ∈ {𝑦 ∣ ∀𝑥 ¬ (𝑦 ≺ 𝑥 ∧ 𝑥 ≺ 𝒫 𝑦)}))
5		breq1 5089	. . . . . . 7 ⊢ (𝑦 = 𝐴 → (𝑦 ≺ 𝑥 ↔ 𝐴 ≺ 𝑥))
6		pweq 4558	. . . . . . . 8 ⊢ (𝑦 = 𝐴 → 𝒫 𝑦 = 𝒫 𝐴)
7	6	breq2d 5098	. . . . . . 7 ⊢ (𝑦 = 𝐴 → (𝑥 ≺ 𝒫 𝑦 ↔ 𝑥 ≺ 𝒫 𝐴))
8	5, 7	anbi12d 631	. . . . . 6 ⊢ (𝑦 = 𝐴 → ((𝑦 ≺ 𝑥 ∧ 𝑥 ≺ 𝒫 𝑦) ↔ (𝐴 ≺ 𝑥 ∧ 𝑥 ≺ 𝒫 𝐴)))
9	8	notbid 317	. . . . 5 ⊢ (𝑦 = 𝐴 → (¬ (𝑦 ≺ 𝑥 ∧ 𝑥 ≺ 𝒫 𝑦) ↔ ¬ (𝐴 ≺ 𝑥 ∧ 𝑥 ≺ 𝒫 𝐴)))
10	9	albidv 1922	. . . 4 ⊢ (𝑦 = 𝐴 → (∀𝑥 ¬ (𝑦 ≺ 𝑥 ∧ 𝑥 ≺ 𝒫 𝑦) ↔ ∀𝑥 ¬ (𝐴 ≺ 𝑥 ∧ 𝑥 ≺ 𝒫 𝐴)))
11	10	elabg 3616	. . 3 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ {𝑦 ∣ ∀𝑥 ¬ (𝑦 ≺ 𝑥 ∧ 𝑥 ≺ 𝒫 𝑦)} ↔ ∀𝑥 ¬ (𝐴 ≺ 𝑥 ∧ 𝑥 ≺ 𝒫 𝐴)))
12	11	orbi2d 913	. 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 282	1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ GCH ↔ (𝐴 ∈ Fin ∨ ∀𝑥 ¬ (𝐴 ≺ 𝑥 ∧ 𝑥 ≺ 𝒫 𝐴))))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 396   ∨ wo 844  ∀wal 1538   = wceq 1540   ∈ wcel 2105  {cab 2713   ∪ cun 3894  𝒫 cpw 4544   class class class wbr 5086   ≺ csdm 8781  Fincfn 8782  GCHcgch 10455

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2707

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-sb 2067  df-clab 2714  df-cleq 2728  df-clel 2814  df-rab 3404  df-v 3442  df-dif 3899  df-un 3901  df-in 3903  df-ss 3913  df-nul 4267  df-if 4471  df-pw 4546  df-sn 4571  df-pr 4573  df-op 4577  df-br 5087  df-gch 10456

This theorem is referenced by:  gchi  10459  engch  10463  hargch  10508  alephgch  10509