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Theorem isfin6 9285
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 9275 . . 3 FinVI = {𝑥 ∣ (𝑥 ≺ 2𝑜𝑥 ≺ (𝑥 × 𝑥))}
21eleq2i 2819 . 2 (𝐴 ∈ FinVI𝐴 ∈ {𝑥 ∣ (𝑥 ≺ 2𝑜𝑥 ≺ (𝑥 × 𝑥))})
3 relsdom 8116 . . . . 5 Rel ≺
43brrelexi 5303 . . . 4 (𝐴 ≺ 2𝑜𝐴 ∈ V)
53brrelexi 5303 . . . 4 (𝐴 ≺ (𝐴 × 𝐴) → 𝐴 ∈ V)
64, 5jaoi 393 . . 3 ((𝐴 ≺ 2𝑜𝐴 ≺ (𝐴 × 𝐴)) → 𝐴 ∈ V)
7 breq1 4795 . . . 4 (𝑥 = 𝐴 → (𝑥 ≺ 2𝑜𝐴 ≺ 2𝑜))
8 id 22 . . . . 5 (𝑥 = 𝐴𝑥 = 𝐴)
98sqxpeqd 5286 . . . . 5 (𝑥 = 𝐴 → (𝑥 × 𝑥) = (𝐴 × 𝐴))
108, 9breq12d 4805 . . . 4 (𝑥 = 𝐴 → (𝑥 ≺ (𝑥 × 𝑥) ↔ 𝐴 ≺ (𝐴 × 𝐴)))
117, 10orbi12d 748 . . 3 (𝑥 = 𝐴 → ((𝑥 ≺ 2𝑜𝑥 ≺ (𝑥 × 𝑥)) ↔ (𝐴 ≺ 2𝑜𝐴 ≺ (𝐴 × 𝐴))))
126, 11elab3 3486 . 2 (𝐴 ∈ {𝑥 ∣ (𝑥 ≺ 2𝑜𝑥 ≺ (𝑥 × 𝑥))} ↔ (𝐴 ≺ 2𝑜𝐴 ≺ (𝐴 × 𝐴)))
132, 12bitri 264 1 (𝐴 ∈ FinVI ↔ (𝐴 ≺ 2𝑜𝐴 ≺ (𝐴 × 𝐴)))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wo 382   = wceq 1620  wcel 2127  {cab 2734  Vcvv 3328   class class class wbr 4792   × cxp 5252  2𝑜c2o 7711  csdm 8108  FinVIcfin6 9268
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1859  ax-4 1874  ax-5 1976  ax-6 2042  ax-7 2078  ax-9 2136  ax-10 2156  ax-11 2171  ax-12 2184  ax-13 2379  ax-ext 2728  ax-sep 4921  ax-nul 4929  ax-pr 5043
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1623  df-ex 1842  df-nf 1847  df-sb 2035  df-clab 2735  df-cleq 2741  df-clel 2744  df-nfc 2879  df-ral 3043  df-rex 3044  df-rab 3047  df-v 3330  df-dif 3706  df-un 3708  df-in 3710  df-ss 3717  df-nul 4047  df-if 4219  df-sn 4310  df-pr 4312  df-op 4316  df-br 4793  df-opab 4853  df-xp 5260  df-rel 5261  df-dom 8111  df-sdom 8112  df-fin6 9275
This theorem is referenced by:  fin56  9378  fin67  9380
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