Theorem isfin6 10066

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 10056	. . 3 ⊢ FinVI = {𝑥 ∣ (𝑥 ≺ 2o ∨ 𝑥 ≺ (𝑥 × 𝑥))}
2	1	eleq2i 2830	. 2 ⊢ (𝐴 ∈ FinVI ↔ 𝐴 ∈ {𝑥 ∣ (𝑥 ≺ 2o ∨ 𝑥 ≺ (𝑥 × 𝑥))})
3		relsdom 8727	. . . . 5 ⊢ Rel ≺
4	3	brrelex1i 5638	. . . 4 ⊢ (𝐴 ≺ 2o → 𝐴 ∈ V)
5	3	brrelex1i 5638	. . . 4 ⊢ (𝐴 ≺ (𝐴 × 𝐴) → 𝐴 ∈ V)
6	4, 5	jaoi 854	. . 3 ⊢ ((𝐴 ≺ 2o ∨ 𝐴 ≺ (𝐴 × 𝐴)) → 𝐴 ∈ V)
7		breq1 5076	. . . 4 ⊢ (𝑥 = 𝐴 → (𝑥 ≺ 2o ↔ 𝐴 ≺ 2o))
8		id 22	. . . . 5 ⊢ (𝑥 = 𝐴 → 𝑥 = 𝐴)
9	8	sqxpeqd 5616	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥 × 𝑥) = (𝐴 × 𝐴))
10	8, 9	breq12d 5086	. . . 4 ⊢ (𝑥 = 𝐴 → (𝑥 ≺ (𝑥 × 𝑥) ↔ 𝐴 ≺ (𝐴 × 𝐴)))
11	7, 10	orbi12d 916	. . 3 ⊢ (𝑥 = 𝐴 → ((𝑥 ≺ 2o ∨ 𝑥 ≺ (𝑥 × 𝑥)) ↔ (𝐴 ≺ 2o ∨ 𝐴 ≺ (𝐴 × 𝐴))))
12	6, 11	elab3 3616	. 2 ⊢ (𝐴 ∈ {𝑥 ∣ (𝑥 ≺ 2o ∨ 𝑥 ≺ (𝑥 × 𝑥))} ↔ (𝐴 ≺ 2o ∨ 𝐴 ≺ (𝐴 × 𝐴)))
13	2, 12	bitri 274	1 ⊢ (𝐴 ∈ FinVI ↔ (𝐴 ≺ 2o ∨ 𝐴 ≺ (𝐴 × 𝐴)))

Syntax hints:   ↔ wb 205   ∨ wo 844   = wceq 1539   ∈ wcel 2106  {cab 2715  Vcvv 3429   class class class wbr 5073   × cxp 5582  2_oc2o 8278   ≺ csdm 8719  Fin^VIcfin6 10049

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709  ax-sep 5221  ax-nul 5228  ax-pr 5350

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3431  df-dif 3889  df-un 3891  df-in 3893  df-ss 3903  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-br 5074  df-opab 5136  df-xp 5590  df-rel 5591  df-dom 8722  df-sdom 8723  df-fin6 10056

This theorem is referenced by:  fin56  10159  fin67  10161