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Theorem isfin5 10339
Description: Definition of a V-finite set. (Contributed by Stefan O'Rear, 16-May-2015.)
Assertion
Ref Expression
isfin5 (𝐴 ∈ FinV ↔ (𝐴 = ∅ ∨ 𝐴 ≺ (𝐴𝐴)))

Proof of Theorem isfin5
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 df-fin5 10329 . . 3 FinV = {𝑥 ∣ (𝑥 = ∅ ∨ 𝑥 ≺ (𝑥𝑥))}
21eleq2i 2833 . 2 (𝐴 ∈ FinV𝐴 ∈ {𝑥 ∣ (𝑥 = ∅ ∨ 𝑥 ≺ (𝑥𝑥))})
3 id 22 . . . . 5 (𝐴 = ∅ → 𝐴 = ∅)
4 0ex 5307 . . . . 5 ∅ ∈ V
53, 4eqeltrdi 2849 . . . 4 (𝐴 = ∅ → 𝐴 ∈ V)
6 relsdom 8992 . . . . 5 Rel ≺
76brrelex1i 5741 . . . 4 (𝐴 ≺ (𝐴𝐴) → 𝐴 ∈ V)
85, 7jaoi 858 . . 3 ((𝐴 = ∅ ∨ 𝐴 ≺ (𝐴𝐴)) → 𝐴 ∈ V)
9 eqeq1 2741 . . . 4 (𝑥 = 𝐴 → (𝑥 = ∅ ↔ 𝐴 = ∅))
10 id 22 . . . . 5 (𝑥 = 𝐴𝑥 = 𝐴)
11 djueq12 9944 . . . . . 6 ((𝑥 = 𝐴𝑥 = 𝐴) → (𝑥𝑥) = (𝐴𝐴))
1211anidms 566 . . . . 5 (𝑥 = 𝐴 → (𝑥𝑥) = (𝐴𝐴))
1310, 12breq12d 5156 . . . 4 (𝑥 = 𝐴 → (𝑥 ≺ (𝑥𝑥) ↔ 𝐴 ≺ (𝐴𝐴)))
149, 13orbi12d 919 . . 3 (𝑥 = 𝐴 → ((𝑥 = ∅ ∨ 𝑥 ≺ (𝑥𝑥)) ↔ (𝐴 = ∅ ∨ 𝐴 ≺ (𝐴𝐴))))
158, 14elab3 3686 . 2 (𝐴 ∈ {𝑥 ∣ (𝑥 = ∅ ∨ 𝑥 ≺ (𝑥𝑥))} ↔ (𝐴 = ∅ ∨ 𝐴 ≺ (𝐴𝐴)))
162, 15bitri 275 1 (𝐴 ∈ FinV ↔ (𝐴 = ∅ ∨ 𝐴 ≺ (𝐴𝐴)))
Colors of variables: wff setvar class
Syntax hints:  wb 206  wo 848   = wceq 1540  wcel 2108  {cab 2714  Vcvv 3480  c0 4333   class class class wbr 5143  csdm 8984  cdju 9938  FinVcfin5 10322
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 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-xp 5691  df-rel 5692  df-dom 8987  df-sdom 8988  df-dju 9941  df-fin5 10329
This theorem is referenced by:  isfin5-2  10431  fin56  10433
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