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Theorem isfin5 9713
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 9703 . . 3 FinV = {𝑥 ∣ (𝑥 = ∅ ∨ 𝑥 ≺ (𝑥𝑥))}
21eleq2i 2902 . 2 (𝐴 ∈ FinV𝐴 ∈ {𝑥 ∣ (𝑥 = ∅ ∨ 𝑥 ≺ (𝑥𝑥))})
3 id 22 . . . . 5 (𝐴 = ∅ → 𝐴 = ∅)
4 0ex 5202 . . . . 5 ∅ ∈ V
53, 4syl6eqel 2919 . . . 4 (𝐴 = ∅ → 𝐴 ∈ V)
6 relsdom 8508 . . . . 5 Rel ≺
76brrelex1i 5601 . . . 4 (𝐴 ≺ (𝐴𝐴) → 𝐴 ∈ V)
85, 7jaoi 853 . . 3 ((𝐴 = ∅ ∨ 𝐴 ≺ (𝐴𝐴)) → 𝐴 ∈ V)
9 eqeq1 2823 . . . 4 (𝑥 = 𝐴 → (𝑥 = ∅ ↔ 𝐴 = ∅))
10 id 22 . . . . 5 (𝑥 = 𝐴𝑥 = 𝐴)
11 djueq12 9325 . . . . . 6 ((𝑥 = 𝐴𝑥 = 𝐴) → (𝑥𝑥) = (𝐴𝐴))
1211anidms 569 . . . . 5 (𝑥 = 𝐴 → (𝑥𝑥) = (𝐴𝐴))
1310, 12breq12d 5070 . . . 4 (𝑥 = 𝐴 → (𝑥 ≺ (𝑥𝑥) ↔ 𝐴 ≺ (𝐴𝐴)))
149, 13orbi12d 915 . . 3 (𝑥 = 𝐴 → ((𝑥 = ∅ ∨ 𝑥 ≺ (𝑥𝑥)) ↔ (𝐴 = ∅ ∨ 𝐴 ≺ (𝐴𝐴))))
158, 14elab3 3672 . 2 (𝐴 ∈ {𝑥 ∣ (𝑥 = ∅ ∨ 𝑥 ≺ (𝑥𝑥))} ↔ (𝐴 = ∅ ∨ 𝐴 ≺ (𝐴𝐴)))
162, 15bitri 277 1 (𝐴 ∈ FinV ↔ (𝐴 = ∅ ∨ 𝐴 ≺ (𝐴𝐴)))
Colors of variables: wff setvar class
Syntax hints:  wb 208  wo 843   = wceq 1531  wcel 2108  {cab 2797  Vcvv 3493  c0 4289   class class class wbr 5057  csdm 8500  cdju 9319  FinVcfin5 9696
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1905  ax-6 1964  ax-7 2009  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2154  ax-12 2170  ax-ext 2791  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1084  df-tru 1534  df-ex 1775  df-nf 1779  df-sb 2064  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ral 3141  df-rex 3142  df-rab 3145  df-v 3495  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-nul 4290  df-if 4466  df-sn 4560  df-pr 4562  df-op 4566  df-br 5058  df-opab 5120  df-xp 5554  df-rel 5555  df-dom 8503  df-sdom 8504  df-dju 9322  df-fin5 9703
This theorem is referenced by:  isfin5-2  9805  fin56  9807
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