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Theorem isfin6 9067
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 9057 . . 3 FinVI = {𝑥 ∣ (𝑥 ≺ 2𝑜𝑥 ≺ (𝑥 × 𝑥))}
21eleq2i 2696 . 2 (𝐴 ∈ FinVI𝐴 ∈ {𝑥 ∣ (𝑥 ≺ 2𝑜𝑥 ≺ (𝑥 × 𝑥))})
3 relsdom 7907 . . . . 5 Rel ≺
43brrelexi 5123 . . . 4 (𝐴 ≺ 2𝑜𝐴 ∈ V)
53brrelexi 5123 . . . 4 (𝐴 ≺ (𝐴 × 𝐴) → 𝐴 ∈ V)
64, 5jaoi 394 . . 3 ((𝐴 ≺ 2𝑜𝐴 ≺ (𝐴 × 𝐴)) → 𝐴 ∈ V)
7 breq1 4621 . . . 4 (𝑥 = 𝐴 → (𝑥 ≺ 2𝑜𝐴 ≺ 2𝑜))
8 id 22 . . . . 5 (𝑥 = 𝐴𝑥 = 𝐴)
98sqxpeqd 5106 . . . . 5 (𝑥 = 𝐴 → (𝑥 × 𝑥) = (𝐴 × 𝐴))
108, 9breq12d 4631 . . . 4 (𝑥 = 𝐴 → (𝑥 ≺ (𝑥 × 𝑥) ↔ 𝐴 ≺ (𝐴 × 𝐴)))
117, 10orbi12d 745 . . 3 (𝑥 = 𝐴 → ((𝑥 ≺ 2𝑜𝑥 ≺ (𝑥 × 𝑥)) ↔ (𝐴 ≺ 2𝑜𝐴 ≺ (𝐴 × 𝐴))))
126, 11elab3 3346 . 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 383   = wceq 1480  wcel 1992  {cab 2612  Vcvv 3191   class class class wbr 4618   × cxp 5077  2𝑜c2o 7500  csdm 7899  FinVIcfin6 9050
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-sep 4746  ax-nul 4754  ax-pr 4872
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3193  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3897  df-if 4064  df-sn 4154  df-pr 4156  df-op 4160  df-br 4619  df-opab 4679  df-xp 5085  df-rel 5086  df-dom 7902  df-sdom 7903  df-fin6 9057
This theorem is referenced by:  fin56  9160  fin67  9162
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