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Theorem isfin6 10341
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 10331 . . 3 FinVI = {𝑥 ∣ (𝑥 ≺ 2o𝑥 ≺ (𝑥 × 𝑥))}
21eleq2i 2832 . 2 (𝐴 ∈ FinVI𝐴 ∈ {𝑥 ∣ (𝑥 ≺ 2o𝑥 ≺ (𝑥 × 𝑥))})
3 relsdom 8993 . . . . 5 Rel ≺
43brrelex1i 5740 . . . 4 (𝐴 ≺ 2o𝐴 ∈ V)
53brrelex1i 5740 . . . 4 (𝐴 ≺ (𝐴 × 𝐴) → 𝐴 ∈ V)
64, 5jaoi 857 . . 3 ((𝐴 ≺ 2o𝐴 ≺ (𝐴 × 𝐴)) → 𝐴 ∈ V)
7 breq1 5145 . . . 4 (𝑥 = 𝐴 → (𝑥 ≺ 2o𝐴 ≺ 2o))
8 id 22 . . . . 5 (𝑥 = 𝐴𝑥 = 𝐴)
98sqxpeqd 5716 . . . . 5 (𝑥 = 𝐴 → (𝑥 × 𝑥) = (𝐴 × 𝐴))
108, 9breq12d 5155 . . . 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 3685 . 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 1539  wcel 2107  {cab 2713  Vcvv 3479   class class class wbr 5142   × cxp 5682  2oc2o 8501  csdm 8985  FinVIcfin6 10324
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-br 5143  df-opab 5205  df-xp 5690  df-rel 5691  df-dom 8988  df-sdom 8989  df-fin6 10331
This theorem is referenced by:  fin56  10434  fin67  10436
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