Theorem isfin6 9160

Description: Definition of a VI-finite set. (Contributed by Stefan O'Rear, 16-May-2015.)

Assertion

Ref	Expression
isfin6	⊢ (𝐴 ∈ FinVI ↔ (𝐴 ≺ 2𝑜 ∨ 𝐴 ≺ (𝐴 × 𝐴)))

Proof of Theorem isfin6

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-fin6 9150	. . 3 ⊢ FinVI = {𝑥 ∣ (𝑥 ≺ 2𝑜 ∨ 𝑥 ≺ (𝑥 × 𝑥))}
2	1	eleq2i 2722	. 2 ⊢ (𝐴 ∈ FinVI ↔ 𝐴 ∈ {𝑥 ∣ (𝑥 ≺ 2𝑜 ∨ 𝑥 ≺ (𝑥 × 𝑥))})
3		relsdom 8004	. . . . 5 ⊢ Rel ≺
4	3	brrelexi 5192	. . . 4 ⊢ (𝐴 ≺ 2𝑜 → 𝐴 ∈ V)
5	3	brrelexi 5192	. . . 4 ⊢ (𝐴 ≺ (𝐴 × 𝐴) → 𝐴 ∈ V)
6	4, 5	jaoi 393	. . 3 ⊢ ((𝐴 ≺ 2𝑜 ∨ 𝐴 ≺ (𝐴 × 𝐴)) → 𝐴 ∈ V)
7		breq1 4688	. . . 4 ⊢ (𝑥 = 𝐴 → (𝑥 ≺ 2𝑜 ↔ 𝐴 ≺ 2𝑜))
8		id 22	. . . . 5 ⊢ (𝑥 = 𝐴 → 𝑥 = 𝐴)
9	8	sqxpeqd 5175	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥 × 𝑥) = (𝐴 × 𝐴))
10	8, 9	breq12d 4698	. . . 4 ⊢ (𝑥 = 𝐴 → (𝑥 ≺ (𝑥 × 𝑥) ↔ 𝐴 ≺ (𝐴 × 𝐴)))
11	7, 10	orbi12d 746	. . 3 ⊢ (𝑥 = 𝐴 → ((𝑥 ≺ 2𝑜 ∨ 𝑥 ≺ (𝑥 × 𝑥)) ↔ (𝐴 ≺ 2𝑜 ∨ 𝐴 ≺ (𝐴 × 𝐴))))
12	6, 11	elab3 3390	. 2 ⊢ (𝐴 ∈ {𝑥 ∣ (𝑥 ≺ 2𝑜 ∨ 𝑥 ≺ (𝑥 × 𝑥))} ↔ (𝐴 ≺ 2𝑜 ∨ 𝐴 ≺ (𝐴 × 𝐴)))
13	2, 12	bitri 264	1 ⊢ (𝐴 ∈ FinVI ↔ (𝐴 ≺ 2𝑜 ∨ 𝐴 ≺ (𝐴 × 𝐴)))

Syntax hints:   ↔ wb 196   ∨ wo 382   = wceq 1523   ∈ wcel 2030  {cab 2637  Vcvv 3231   class class class wbr 4685   × cxp 5141  2_𝑜c2o 7599   ≺ csdm 7996  Fin^VIcfin6 9143

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-br 4686  df-opab 4746  df-xp 5149  df-rel 5150  df-dom 7999  df-sdom 8000  df-fin6 9150

This theorem is referenced by:  fin56  9253  fin67  9255