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Theorem isfin6 10297
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 10287 . . 3 FinVI = {𝑥 ∣ (𝑥 ≺ 2o𝑥 ≺ (𝑥 × 𝑥))}
21eleq2i 2823 . 2 (𝐴 ∈ FinVI𝐴 ∈ {𝑥 ∣ (𝑥 ≺ 2o𝑥 ≺ (𝑥 × 𝑥))})
3 relsdom 8948 . . . . 5 Rel ≺
43brrelex1i 5731 . . . 4 (𝐴 ≺ 2o𝐴 ∈ V)
53brrelex1i 5731 . . . 4 (𝐴 ≺ (𝐴 × 𝐴) → 𝐴 ∈ V)
64, 5jaoi 853 . . 3 ((𝐴 ≺ 2o𝐴 ≺ (𝐴 × 𝐴)) → 𝐴 ∈ V)
7 breq1 5150 . . . 4 (𝑥 = 𝐴 → (𝑥 ≺ 2o𝐴 ≺ 2o))
8 id 22 . . . . 5 (𝑥 = 𝐴𝑥 = 𝐴)
98sqxpeqd 5707 . . . . 5 (𝑥 = 𝐴 → (𝑥 × 𝑥) = (𝐴 × 𝐴))
108, 9breq12d 5160 . . . 4 (𝑥 = 𝐴 → (𝑥 ≺ (𝑥 × 𝑥) ↔ 𝐴 ≺ (𝐴 × 𝐴)))
117, 10orbi12d 915 . . 3 (𝑥 = 𝐴 → ((𝑥 ≺ 2o𝑥 ≺ (𝑥 × 𝑥)) ↔ (𝐴 ≺ 2o𝐴 ≺ (𝐴 × 𝐴))))
126, 11elab3 3675 . 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 843   = wceq 1539  wcel 2104  {cab 2707  Vcvv 3472   class class class wbr 5147   × cxp 5673  2oc2o 8462  csdm 8940  FinVIcfin6 10280
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-ext 2701  ax-sep 5298  ax-nul 5305  ax-pr 5426
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2722  df-clel 2808  df-ral 3060  df-rex 3069  df-rab 3431  df-v 3474  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-br 5148  df-opab 5210  df-xp 5681  df-rel 5682  df-dom 8943  df-sdom 8944  df-fin6 10287
This theorem is referenced by:  fin56  10390  fin67  10392
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