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Theorem isfin5 9004
Description: Definition of a V-finite set. (Contributed by Stefan O'Rear, 16-May-2015.)
Assertion
Ref Expression
isfin5 (𝐴 ∈ FinV ↔ (𝐴 = ∅ ∨ 𝐴 ≺ (𝐴 +𝑐 𝐴)))

Proof of Theorem isfin5
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 df-fin5 8994 . . 3 FinV = {𝑥 ∣ (𝑥 = ∅ ∨ 𝑥 ≺ (𝑥 +𝑐 𝑥))}
21eleq2i 2680 . 2 (𝐴 ∈ FinV𝐴 ∈ {𝑥 ∣ (𝑥 = ∅ ∨ 𝑥 ≺ (𝑥 +𝑐 𝑥))})
3 id 22 . . . . 5 (𝐴 = ∅ → 𝐴 = ∅)
4 0ex 4718 . . . . 5 ∅ ∈ V
53, 4syl6eqel 2696 . . . 4 (𝐴 = ∅ → 𝐴 ∈ V)
6 relsdom 7848 . . . . 5 Rel ≺
76brrelexi 5082 . . . 4 (𝐴 ≺ (𝐴 +𝑐 𝐴) → 𝐴 ∈ V)
85, 7jaoi 393 . . 3 ((𝐴 = ∅ ∨ 𝐴 ≺ (𝐴 +𝑐 𝐴)) → 𝐴 ∈ V)
9 eqeq1 2614 . . . 4 (𝑥 = 𝐴 → (𝑥 = ∅ ↔ 𝐴 = ∅))
10 id 22 . . . . 5 (𝑥 = 𝐴𝑥 = 𝐴)
1110, 10oveq12d 6567 . . . . 5 (𝑥 = 𝐴 → (𝑥 +𝑐 𝑥) = (𝐴 +𝑐 𝐴))
1210, 11breq12d 4596 . . . 4 (𝑥 = 𝐴 → (𝑥 ≺ (𝑥 +𝑐 𝑥) ↔ 𝐴 ≺ (𝐴 +𝑐 𝐴)))
139, 12orbi12d 742 . . 3 (𝑥 = 𝐴 → ((𝑥 = ∅ ∨ 𝑥 ≺ (𝑥 +𝑐 𝑥)) ↔ (𝐴 = ∅ ∨ 𝐴 ≺ (𝐴 +𝑐 𝐴))))
148, 13elab3 3327 . 2 (𝐴 ∈ {𝑥 ∣ (𝑥 = ∅ ∨ 𝑥 ≺ (𝑥 +𝑐 𝑥))} ↔ (𝐴 = ∅ ∨ 𝐴 ≺ (𝐴 +𝑐 𝐴)))
152, 14bitri 263 1 (𝐴 ∈ FinV ↔ (𝐴 = ∅ ∨ 𝐴 ≺ (𝐴 +𝑐 𝐴)))
Colors of variables: wff setvar class
Syntax hints:  wb 195  wo 382   = wceq 1475  wcel 1977  {cab 2596  Vcvv 3173  c0 3874   class class class wbr 4583  (class class class)co 6549  csdm 7840   +𝑐 ccda 8872  FinVcfin5 8987
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-xp 5044  df-rel 5045  df-iota 5768  df-fv 5812  df-ov 6552  df-dom 7843  df-sdom 7844  df-fin5 8994
This theorem is referenced by:  isfin5-2  9096  fin56  9098
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