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Theorem isfin5 9709
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 9699 . . 3 FinV = {𝑥 ∣ (𝑥 = ∅ ∨ 𝑥 ≺ (𝑥𝑥))}
21eleq2i 2901 . 2 (𝐴 ∈ FinV𝐴 ∈ {𝑥 ∣ (𝑥 = ∅ ∨ 𝑥 ≺ (𝑥𝑥))})
3 id 22 . . . . 5 (𝐴 = ∅ → 𝐴 = ∅)
4 0ex 5202 . . . . 5 ∅ ∈ V
53, 4syl6eqel 2918 . . . 4 (𝐴 = ∅ → 𝐴 ∈ V)
6 relsdom 8504 . . . . 5 Rel ≺
76brrelex1i 5601 . . . 4 (𝐴 ≺ (𝐴𝐴) → 𝐴 ∈ V)
85, 7jaoi 851 . . 3 ((𝐴 = ∅ ∨ 𝐴 ≺ (𝐴𝐴)) → 𝐴 ∈ V)
9 eqeq1 2822 . . . 4 (𝑥 = 𝐴 → (𝑥 = ∅ ↔ 𝐴 = ∅))
10 id 22 . . . . 5 (𝑥 = 𝐴𝑥 = 𝐴)
11 djueq12 9321 . . . . . 6 ((𝑥 = 𝐴𝑥 = 𝐴) → (𝑥𝑥) = (𝐴𝐴))
1211anidms 567 . . . . 5 (𝑥 = 𝐴 → (𝑥𝑥) = (𝐴𝐴))
1310, 12breq12d 5070 . . . 4 (𝑥 = 𝐴 → (𝑥 ≺ (𝑥𝑥) ↔ 𝐴 ≺ (𝐴𝐴)))
149, 13orbi12d 912 . . 3 (𝑥 = 𝐴 → ((𝑥 = ∅ ∨ 𝑥 ≺ (𝑥𝑥)) ↔ (𝐴 = ∅ ∨ 𝐴 ≺ (𝐴𝐴))))
158, 14elab3 3671 . 2 (𝐴 ∈ {𝑥 ∣ (𝑥 = ∅ ∨ 𝑥 ≺ (𝑥𝑥))} ↔ (𝐴 = ∅ ∨ 𝐴 ≺ (𝐴𝐴)))
162, 15bitri 276 1 (𝐴 ∈ FinV ↔ (𝐴 = ∅ ∨ 𝐴 ≺ (𝐴𝐴)))
Colors of variables: wff setvar class
Syntax hints:  wb 207  wo 841   = wceq 1528  wcel 2105  {cab 2796  Vcvv 3492  c0 4288   class class class wbr 5057  csdm 8496  cdju 9315  FinVcfin5 9692
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-br 5058  df-opab 5120  df-xp 5554  df-rel 5555  df-dom 8499  df-sdom 8500  df-dju 9318  df-fin5 9699
This theorem is referenced by:  isfin5-2  9801  fin56  9803
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