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Theorem isfin6 10338
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 10328 . . 3 FinVI = {𝑥 ∣ (𝑥 ≺ 2o𝑥 ≺ (𝑥 × 𝑥))}
21eleq2i 2831 . 2 (𝐴 ∈ FinVI𝐴 ∈ {𝑥 ∣ (𝑥 ≺ 2o𝑥 ≺ (𝑥 × 𝑥))})
3 relsdom 8991 . . . . 5 Rel ≺
43brrelex1i 5745 . . . 4 (𝐴 ≺ 2o𝐴 ∈ V)
53brrelex1i 5745 . . . 4 (𝐴 ≺ (𝐴 × 𝐴) → 𝐴 ∈ V)
64, 5jaoi 857 . . 3 ((𝐴 ≺ 2o𝐴 ≺ (𝐴 × 𝐴)) → 𝐴 ∈ V)
7 breq1 5151 . . . 4 (𝑥 = 𝐴 → (𝑥 ≺ 2o𝐴 ≺ 2o))
8 id 22 . . . . 5 (𝑥 = 𝐴𝑥 = 𝐴)
98sqxpeqd 5721 . . . . 5 (𝑥 = 𝐴 → (𝑥 × 𝑥) = (𝐴 × 𝐴))
108, 9breq12d 5161 . . . 4 (𝑥 = 𝐴 → (𝑥 ≺ (𝑥 × 𝑥) ↔ 𝐴 ≺ (𝐴 × 𝐴)))
117, 10orbi12d 918 . . 3 (𝑥 = 𝐴 → ((𝑥 ≺ 2o𝑥 ≺ (𝑥 × 𝑥)) ↔ (𝐴 ≺ 2o𝐴 ≺ (𝐴 × 𝐴))))
126, 11elab3 3689 . 2 (𝐴 ∈ {𝑥 ∣ (𝑥 ≺ 2o𝑥 ≺ (𝑥 × 𝑥))} ↔ (𝐴 ≺ 2o𝐴 ≺ (𝐴 × 𝐴)))
132, 12bitri 275 1 (𝐴 ∈ FinVI ↔ (𝐴 ≺ 2o𝐴 ≺ (𝐴 × 𝐴)))
Colors of variables: wff setvar class
Syntax hints:  wb 206  wo 847   = wceq 1537  wcel 2106  {cab 2712  Vcvv 3478   class class class wbr 5148   × cxp 5687  2oc2o 8499  csdm 8983  FinVIcfin6 10321
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-opab 5211  df-xp 5695  df-rel 5696  df-dom 8986  df-sdom 8987  df-fin6 10328
This theorem is referenced by:  fin56  10431  fin67  10433
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