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Theorem isfin6 10339
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 10329 . . 3 FinVI = {𝑥 ∣ (𝑥 ≺ 2o𝑥 ≺ (𝑥 × 𝑥))}
21eleq2i 2817 . 2 (𝐴 ∈ FinVI𝐴 ∈ {𝑥 ∣ (𝑥 ≺ 2o𝑥 ≺ (𝑥 × 𝑥))})
3 relsdom 8980 . . . . 5 Rel ≺
43brrelex1i 5737 . . . 4 (𝐴 ≺ 2o𝐴 ∈ V)
53brrelex1i 5737 . . . 4 (𝐴 ≺ (𝐴 × 𝐴) → 𝐴 ∈ V)
64, 5jaoi 855 . . 3 ((𝐴 ≺ 2o𝐴 ≺ (𝐴 × 𝐴)) → 𝐴 ∈ V)
7 breq1 5155 . . . 4 (𝑥 = 𝐴 → (𝑥 ≺ 2o𝐴 ≺ 2o))
8 id 22 . . . . 5 (𝑥 = 𝐴𝑥 = 𝐴)
98sqxpeqd 5713 . . . . 5 (𝑥 = 𝐴 → (𝑥 × 𝑥) = (𝐴 × 𝐴))
108, 9breq12d 5165 . . . 4 (𝑥 = 𝐴 → (𝑥 ≺ (𝑥 × 𝑥) ↔ 𝐴 ≺ (𝐴 × 𝐴)))
117, 10orbi12d 916 . . 3 (𝑥 = 𝐴 → ((𝑥 ≺ 2o𝑥 ≺ (𝑥 × 𝑥)) ↔ (𝐴 ≺ 2o𝐴 ≺ (𝐴 × 𝐴))))
126, 11elab3 3673 . 2 (𝐴 ∈ {𝑥 ∣ (𝑥 ≺ 2o𝑥 ≺ (𝑥 × 𝑥))} ↔ (𝐴 ≺ 2o𝐴 ≺ (𝐴 × 𝐴)))
132, 12bitri 274 1 (𝐴 ∈ FinVI ↔ (𝐴 ≺ 2o𝐴 ≺ (𝐴 × 𝐴)))
Colors of variables: wff setvar class
Syntax hints:  wb 205  wo 845   = wceq 1533  wcel 2098  {cab 2702  Vcvv 3461   class class class wbr 5152   × cxp 5679  2oc2o 8489  csdm 8972  FinVIcfin6 10322
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2696  ax-sep 5303  ax-nul 5310  ax-pr 5432
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-ral 3051  df-rex 3060  df-rab 3419  df-v 3463  df-dif 3949  df-un 3951  df-ss 3963  df-nul 4325  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-br 5153  df-opab 5215  df-xp 5687  df-rel 5688  df-dom 8975  df-sdom 8976  df-fin6 10329
This theorem is referenced by:  fin56  10432  fin67  10434
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