MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  isfin6 Structured version   Visualization version   GIF version

Theorem isfin6 9410
Description: Definition of a VI-finite set. (Contributed by Stefan O'Rear, 16-May-2015.)
Assertion
Ref Expression
isfin6 (𝐴 ∈ FinVI ↔ (𝐴 ≺ 2𝑜𝐴 ≺ (𝐴 × 𝐴)))

Proof of Theorem isfin6
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 df-fin6 9400 . . 3 FinVI = {𝑥 ∣ (𝑥 ≺ 2𝑜𝑥 ≺ (𝑥 × 𝑥))}
21eleq2i 2870 . 2 (𝐴 ∈ FinVI𝐴 ∈ {𝑥 ∣ (𝑥 ≺ 2𝑜𝑥 ≺ (𝑥 × 𝑥))})
3 relsdom 8202 . . . . 5 Rel ≺
43brrelex1i 5363 . . . 4 (𝐴 ≺ 2𝑜𝐴 ∈ V)
53brrelex1i 5363 . . . 4 (𝐴 ≺ (𝐴 × 𝐴) → 𝐴 ∈ V)
64, 5jaoi 884 . . 3 ((𝐴 ≺ 2𝑜𝐴 ≺ (𝐴 × 𝐴)) → 𝐴 ∈ V)
7 breq1 4846 . . . 4 (𝑥 = 𝐴 → (𝑥 ≺ 2𝑜𝐴 ≺ 2𝑜))
8 id 22 . . . . 5 (𝑥 = 𝐴𝑥 = 𝐴)
98sqxpeqd 5344 . . . . 5 (𝑥 = 𝐴 → (𝑥 × 𝑥) = (𝐴 × 𝐴))
108, 9breq12d 4856 . . . 4 (𝑥 = 𝐴 → (𝑥 ≺ (𝑥 × 𝑥) ↔ 𝐴 ≺ (𝐴 × 𝐴)))
117, 10orbi12d 943 . . 3 (𝑥 = 𝐴 → ((𝑥 ≺ 2𝑜𝑥 ≺ (𝑥 × 𝑥)) ↔ (𝐴 ≺ 2𝑜𝐴 ≺ (𝐴 × 𝐴))))
126, 11elab3 3550 . 2 (𝐴 ∈ {𝑥 ∣ (𝑥 ≺ 2𝑜𝑥 ≺ (𝑥 × 𝑥))} ↔ (𝐴 ≺ 2𝑜𝐴 ≺ (𝐴 × 𝐴)))
132, 12bitri 267 1 (𝐴 ∈ FinVI ↔ (𝐴 ≺ 2𝑜𝐴 ≺ (𝐴 × 𝐴)))
Colors of variables: wff setvar class
Syntax hints:  wb 198  wo 874   = wceq 1653  wcel 2157  {cab 2785  Vcvv 3385   class class class wbr 4843   × cxp 5310  2𝑜c2o 7793  csdm 8194  FinVIcfin6 9393
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-sep 4975  ax-nul 4983  ax-pr 5097
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3387  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-sn 4369  df-pr 4371  df-op 4375  df-br 4844  df-opab 4906  df-xp 5318  df-rel 5319  df-dom 8197  df-sdom 8198  df-fin6 9400
This theorem is referenced by:  fin56  9503  fin67  9505
  Copyright terms: Public domain W3C validator