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Theorem isfin5 9159
 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 9149 . . 3 FinV = {𝑥 ∣ (𝑥 = ∅ ∨ 𝑥 ≺ (𝑥 +𝑐 𝑥))}
21eleq2i 2722 . 2 (𝐴 ∈ FinV𝐴 ∈ {𝑥 ∣ (𝑥 = ∅ ∨ 𝑥 ≺ (𝑥 +𝑐 𝑥))})
3 id 22 . . . . 5 (𝐴 = ∅ → 𝐴 = ∅)
4 0ex 4823 . . . . 5 ∅ ∈ V
53, 4syl6eqel 2738 . . . 4 (𝐴 = ∅ → 𝐴 ∈ V)
6 relsdom 8004 . . . . 5 Rel ≺
76brrelexi 5192 . . . 4 (𝐴 ≺ (𝐴 +𝑐 𝐴) → 𝐴 ∈ V)
85, 7jaoi 393 . . 3 ((𝐴 = ∅ ∨ 𝐴 ≺ (𝐴 +𝑐 𝐴)) → 𝐴 ∈ V)
9 eqeq1 2655 . . . 4 (𝑥 = 𝐴 → (𝑥 = ∅ ↔ 𝐴 = ∅))
10 id 22 . . . . 5 (𝑥 = 𝐴𝑥 = 𝐴)
1110, 10oveq12d 6708 . . . . 5 (𝑥 = 𝐴 → (𝑥 +𝑐 𝑥) = (𝐴 +𝑐 𝐴))
1210, 11breq12d 4698 . . . 4 (𝑥 = 𝐴 → (𝑥 ≺ (𝑥 +𝑐 𝑥) ↔ 𝐴 ≺ (𝐴 +𝑐 𝐴)))
139, 12orbi12d 746 . . 3 (𝑥 = 𝐴 → ((𝑥 = ∅ ∨ 𝑥 ≺ (𝑥 +𝑐 𝑥)) ↔ (𝐴 = ∅ ∨ 𝐴 ≺ (𝐴 +𝑐 𝐴))))
148, 13elab3 3390 . 2 (𝐴 ∈ {𝑥 ∣ (𝑥 = ∅ ∨ 𝑥 ≺ (𝑥 +𝑐 𝑥))} ↔ (𝐴 = ∅ ∨ 𝐴 ≺ (𝐴 +𝑐 𝐴)))
152, 14bitri 264 1 (𝐴 ∈ FinV ↔ (𝐴 = ∅ ∨ 𝐴 ≺ (𝐴 +𝑐 𝐴)))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 196   ∨ wo 382   = wceq 1523   ∈ wcel 2030  {cab 2637  Vcvv 3231  ∅c0 3948   class class class wbr 4685  (class class class)co 6690   ≺ csdm 7996   +𝑐 ccda 9027  FinVcfin5 9142 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-xp 5149  df-rel 5150  df-iota 5889  df-fv 5934  df-ov 6693  df-dom 7999  df-sdom 8000  df-fin5 9149 This theorem is referenced by:  isfin5-2  9251  fin56  9253
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