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.

Step	Hyp	Ref	Expression
1		df-fin5 9149	. . 3 ⊢ FinV = {𝑥 ∣ (𝑥 = ∅ ∨ 𝑥 ≺ (𝑥 +𝑐 𝑥))}
2	1	eleq2i 2722	. 2 ⊢ (𝐴 ∈ FinV ↔ 𝐴 ∈ {𝑥 ∣ (𝑥 = ∅ ∨ 𝑥 ≺ (𝑥 +𝑐 𝑥))})
3		id 22	. . . . 5 ⊢ (𝐴 = ∅ → 𝐴 = ∅)
4		0ex 4823	. . . . 5 ⊢ ∅ ∈ V
5	3, 4	syl6eqel 2738	. . . 4 ⊢ (𝐴 = ∅ → 𝐴 ∈ V)
6		relsdom 8004	. . . . 5 ⊢ Rel ≺
7	6	brrelexi 5192	. . . 4 ⊢ (𝐴 ≺ (𝐴 +𝑐 𝐴) → 𝐴 ∈ V)
8	5, 7	jaoi 393	. . 3 ⊢ ((𝐴 = ∅ ∨ 𝐴 ≺ (𝐴 +𝑐 𝐴)) → 𝐴 ∈ V)
9		eqeq1 2655	. . . 4 ⊢ (𝑥 = 𝐴 → (𝑥 = ∅ ↔ 𝐴 = ∅))
10		id 22	. . . . 5 ⊢ (𝑥 = 𝐴 → 𝑥 = 𝐴)
11	10, 10	oveq12d 6708	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥 +𝑐 𝑥) = (𝐴 +𝑐 𝐴))
12	10, 11	breq12d 4698	. . . 4 ⊢ (𝑥 = 𝐴 → (𝑥 ≺ (𝑥 +𝑐 𝑥) ↔ 𝐴 ≺ (𝐴 +𝑐 𝐴)))
13	9, 12	orbi12d 746	. . 3 ⊢ (𝑥 = 𝐴 → ((𝑥 = ∅ ∨ 𝑥 ≺ (𝑥 +𝑐 𝑥)) ↔ (𝐴 = ∅ ∨ 𝐴 ≺ (𝐴 +𝑐 𝐴))))
14	8, 13	elab3 3390	. 2 ⊢ (𝐴 ∈ {𝑥 ∣ (𝑥 = ∅ ∨ 𝑥 ≺ (𝑥 +𝑐 𝑥))} ↔ (𝐴 = ∅ ∨ 𝐴 ≺ (𝐴 +𝑐 𝐴)))
15	2, 14	bitri 264	1 ⊢ (𝐴 ∈ FinV ↔ (𝐴 = ∅ ∨ 𝐴 ≺ (𝐴 +𝑐 𝐴)))

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  Fin^Vcfin5 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