Theorem isfin6 9005

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 8995	. . 3 ⊢ FinVI = {𝑥 ∣ (𝑥 ≺ 2𝑜 ∨ 𝑥 ≺ (𝑥 × 𝑥))}
2	1	eleq2i 2680	. 2 ⊢ (𝐴 ∈ FinVI ↔ 𝐴 ∈ {𝑥 ∣ (𝑥 ≺ 2𝑜 ∨ 𝑥 ≺ (𝑥 × 𝑥))})
3		relsdom 7848	. . . . 5 ⊢ Rel ≺
4	3	brrelexi 5082	. . . 4 ⊢ (𝐴 ≺ 2𝑜 → 𝐴 ∈ V)
5	3	brrelexi 5082	. . . 4 ⊢ (𝐴 ≺ (𝐴 × 𝐴) → 𝐴 ∈ V)
6	4, 5	jaoi 393	. . 3 ⊢ ((𝐴 ≺ 2𝑜 ∨ 𝐴 ≺ (𝐴 × 𝐴)) → 𝐴 ∈ V)
7		breq1 4586	. . . 4 ⊢ (𝑥 = 𝐴 → (𝑥 ≺ 2𝑜 ↔ 𝐴 ≺ 2𝑜))
8		id 22	. . . . 5 ⊢ (𝑥 = 𝐴 → 𝑥 = 𝐴)
9	8	sqxpeqd 5065	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥 × 𝑥) = (𝐴 × 𝐴))
10	8, 9	breq12d 4596	. . . 4 ⊢ (𝑥 = 𝐴 → (𝑥 ≺ (𝑥 × 𝑥) ↔ 𝐴 ≺ (𝐴 × 𝐴)))
11	7, 10	orbi12d 742	. . 3 ⊢ (𝑥 = 𝐴 → ((𝑥 ≺ 2𝑜 ∨ 𝑥 ≺ (𝑥 × 𝑥)) ↔ (𝐴 ≺ 2𝑜 ∨ 𝐴 ≺ (𝐴 × 𝐴))))
12	6, 11	elab3 3327	. 2 ⊢ (𝐴 ∈ {𝑥 ∣ (𝑥 ≺ 2𝑜 ∨ 𝑥 ≺ (𝑥 × 𝑥))} ↔ (𝐴 ≺ 2𝑜 ∨ 𝐴 ≺ (𝐴 × 𝐴)))
13	2, 12	bitri 263	1 ⊢ (𝐴 ∈ FinVI ↔ (𝐴 ≺ 2𝑜 ∨ 𝐴 ≺ (𝐴 × 𝐴)))

Syntax hints:   ↔ wb 195   ∨ wo 382   = wceq 1475   ∈ wcel 1977  {cab 2596  Vcvv 3173   class class class wbr 4583   × cxp 5036  2_𝑜c2o 7441   ≺ csdm 7840  Fin^VIcfin6 8988

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-opab 4644  df-xp 5044  df-rel 5045  df-dom 7843  df-sdom 7844  df-fin6 8995

This theorem is referenced by:  fin56  9098  fin67  9100