Theorem isfin6 10191

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 10181	. . 3 ⊢ FinVI = {𝑥 ∣ (𝑥 ≺ 2o ∨ 𝑥 ≺ (𝑥 × 𝑥))}
2	1	eleq2i 2823	. 2 ⊢ (𝐴 ∈ FinVI ↔ 𝐴 ∈ {𝑥 ∣ (𝑥 ≺ 2o ∨ 𝑥 ≺ (𝑥 × 𝑥))})
3		relsdom 8876	. . . . 5 ⊢ Rel ≺
4	3	brrelex1i 5670	. . . 4 ⊢ (𝐴 ≺ 2o → 𝐴 ∈ V)
5	3	brrelex1i 5670	. . . 4 ⊢ (𝐴 ≺ (𝐴 × 𝐴) → 𝐴 ∈ V)
6	4, 5	jaoi 857	. . 3 ⊢ ((𝐴 ≺ 2o ∨ 𝐴 ≺ (𝐴 × 𝐴)) → 𝐴 ∈ V)
7		breq1 5092	. . . 4 ⊢ (𝑥 = 𝐴 → (𝑥 ≺ 2o ↔ 𝐴 ≺ 2o))
8		id 22	. . . . 5 ⊢ (𝑥 = 𝐴 → 𝑥 = 𝐴)
9	8	sqxpeqd 5646	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥 × 𝑥) = (𝐴 × 𝐴))
10	8, 9	breq12d 5102	. . . 4 ⊢ (𝑥 = 𝐴 → (𝑥 ≺ (𝑥 × 𝑥) ↔ 𝐴 ≺ (𝐴 × 𝐴)))
11	7, 10	orbi12d 918	. . 3 ⊢ (𝑥 = 𝐴 → ((𝑥 ≺ 2o ∨ 𝑥 ≺ (𝑥 × 𝑥)) ↔ (𝐴 ≺ 2o ∨ 𝐴 ≺ (𝐴 × 𝐴))))
12	6, 11	elab3 3637	. 2 ⊢ (𝐴 ∈ {𝑥 ∣ (𝑥 ≺ 2o ∨ 𝑥 ≺ (𝑥 × 𝑥))} ↔ (𝐴 ≺ 2o ∨ 𝐴 ≺ (𝐴 × 𝐴)))
13	2, 12	bitri 275	1 ⊢ (𝐴 ∈ FinVI ↔ (𝐴 ≺ 2o ∨ 𝐴 ≺ (𝐴 × 𝐴)))

Syntax hints:   ↔ wb 206   ∨ wo 847   = wceq 1541   ∈ wcel 2111  {cab 2709  Vcvv 3436   class class class wbr 5089   × cxp 5612  2_oc2o 8379   ≺ csdm 8868  Fin^VIcfin6 10174

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 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703  ax-sep 5232  ax-nul 5242  ax-pr 5368

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-ss 3914  df-nul 4281  df-if 4473  df-sn 4574  df-pr 4576  df-op 4580  df-br 5090  df-opab 5152  df-xp 5620  df-rel 5621  df-dom 8871  df-sdom 8872  df-fin6 10181

This theorem is referenced by:  fin56  10284  fin67  10286