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Theorem isfin6 9721
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 9711 . . 3 FinVI = {𝑥 ∣ (𝑥 ≺ 2o𝑥 ≺ (𝑥 × 𝑥))}
21eleq2i 2904 . 2 (𝐴 ∈ FinVI𝐴 ∈ {𝑥 ∣ (𝑥 ≺ 2o𝑥 ≺ (𝑥 × 𝑥))})
3 relsdom 8515 . . . . 5 Rel ≺
43brrelex1i 5607 . . . 4 (𝐴 ≺ 2o𝐴 ∈ V)
53brrelex1i 5607 . . . 4 (𝐴 ≺ (𝐴 × 𝐴) → 𝐴 ∈ V)
64, 5jaoi 853 . . 3 ((𝐴 ≺ 2o𝐴 ≺ (𝐴 × 𝐴)) → 𝐴 ∈ V)
7 breq1 5068 . . . 4 (𝑥 = 𝐴 → (𝑥 ≺ 2o𝐴 ≺ 2o))
8 id 22 . . . . 5 (𝑥 = 𝐴𝑥 = 𝐴)
98sqxpeqd 5586 . . . . 5 (𝑥 = 𝐴 → (𝑥 × 𝑥) = (𝐴 × 𝐴))
108, 9breq12d 5078 . . . 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 3673 . 2 (𝐴 ∈ {𝑥 ∣ (𝑥 ≺ 2o𝑥 ≺ (𝑥 × 𝑥))} ↔ (𝐴 ≺ 2o𝐴 ≺ (𝐴 × 𝐴)))
132, 12bitri 277 1 (𝐴 ∈ FinVI ↔ (𝐴 ≺ 2o𝐴 ≺ (𝐴 × 𝐴)))
Colors of variables: wff setvar class
Syntax hints:  wb 208  wo 843   = wceq 1533  wcel 2110  {cab 2799  Vcvv 3494   class class class wbr 5065   × cxp 5552  2oc2o 8095  csdm 8507  FinVIcfin6 9704
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 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-sep 5202  ax-nul 5209  ax-pr 5329
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-sn 4567  df-pr 4569  df-op 4573  df-br 5066  df-opab 5128  df-xp 5560  df-rel 5561  df-dom 8510  df-sdom 8511  df-fin6 9711
This theorem is referenced by:  fin56  9814  fin67  9816
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