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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 ≺ 2𝑜𝐴 ≺ (𝐴 × 𝐴)))

Proof of Theorem isfin6
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 df-fin6 9150 . . 3 FinVI = {𝑥 ∣ (𝑥 ≺ 2𝑜𝑥 ≺ (𝑥 × 𝑥))}
21eleq2i 2722 . 2 (𝐴 ∈ FinVI𝐴 ∈ {𝑥 ∣ (𝑥 ≺ 2𝑜𝑥 ≺ (𝑥 × 𝑥))})
3 relsdom 8004 . . . . 5 Rel ≺
43brrelexi 5192 . . . 4 (𝐴 ≺ 2𝑜𝐴 ∈ V)
53brrelexi 5192 . . . 4 (𝐴 ≺ (𝐴 × 𝐴) → 𝐴 ∈ V)
64, 5jaoi 393 . . 3 ((𝐴 ≺ 2𝑜𝐴 ≺ (𝐴 × 𝐴)) → 𝐴 ∈ V)
7 breq1 4688 . . . 4 (𝑥 = 𝐴 → (𝑥 ≺ 2𝑜𝐴 ≺ 2𝑜))
8 id 22 . . . . 5 (𝑥 = 𝐴𝑥 = 𝐴)
98sqxpeqd 5175 . . . . 5 (𝑥 = 𝐴 → (𝑥 × 𝑥) = (𝐴 × 𝐴))
108, 9breq12d 4698 . . . 4 (𝑥 = 𝐴 → (𝑥 ≺ (𝑥 × 𝑥) ↔ 𝐴 ≺ (𝐴 × 𝐴)))
117, 10orbi12d 746 . . 3 (𝑥 = 𝐴 → ((𝑥 ≺ 2𝑜𝑥 ≺ (𝑥 × 𝑥)) ↔ (𝐴 ≺ 2𝑜𝐴 ≺ (𝐴 × 𝐴))))
126, 11elab3 3390 . 2 (𝐴 ∈ {𝑥 ∣ (𝑥 ≺ 2𝑜𝑥 ≺ (𝑥 × 𝑥))} ↔ (𝐴 ≺ 2𝑜𝐴 ≺ (𝐴 × 𝐴)))
132, 12bitri 264 1 (𝐴 ∈ FinVI ↔ (𝐴 ≺ 2𝑜𝐴 ≺ (𝐴 × 𝐴)))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wo 382   = wceq 1523  wcel 2030  {cab 2637  Vcvv 3231   class class class wbr 4685   × cxp 5141  2𝑜c2o 7599  csdm 7996  FinVIcfin6 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
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