Theorem isfin5 9710

Description: Definition of a V-finite set. (Contributed by Stefan O'Rear, 16-May-2015.)

Assertion

Ref	Expression
isfin5	⊢ (𝐴 ∈ FinV ↔ (𝐴 = ∅ ∨ 𝐴 ≺ (𝐴 ⊔ 𝐴)))

Proof of Theorem isfin5

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-fin5 9700	. . 3 ⊢ FinV = {𝑥 ∣ (𝑥 = ∅ ∨ 𝑥 ≺ (𝑥 ⊔ 𝑥))}
2	1	eleq2i 2881	. 2 ⊢ (𝐴 ∈ FinV ↔ 𝐴 ∈ {𝑥 ∣ (𝑥 = ∅ ∨ 𝑥 ≺ (𝑥 ⊔ 𝑥))})
3		id 22	. . . . 5 ⊢ (𝐴 = ∅ → 𝐴 = ∅)
4		0ex 5175	. . . . 5 ⊢ ∅ ∈ V
5	3, 4	eqeltrdi 2898	. . . 4 ⊢ (𝐴 = ∅ → 𝐴 ∈ V)
6		relsdom 8499	. . . . 5 ⊢ Rel ≺
7	6	brrelex1i 5572	. . . 4 ⊢ (𝐴 ≺ (𝐴 ⊔ 𝐴) → 𝐴 ∈ V)
8	5, 7	jaoi 854	. . 3 ⊢ ((𝐴 = ∅ ∨ 𝐴 ≺ (𝐴 ⊔ 𝐴)) → 𝐴 ∈ V)
9		eqeq1 2802	. . . 4 ⊢ (𝑥 = 𝐴 → (𝑥 = ∅ ↔ 𝐴 = ∅))
10		id 22	. . . . 5 ⊢ (𝑥 = 𝐴 → 𝑥 = 𝐴)
11		djueq12 9317	. . . . . 6 ⊢ ((𝑥 = 𝐴 ∧ 𝑥 = 𝐴) → (𝑥 ⊔ 𝑥) = (𝐴 ⊔ 𝐴))
12	11	anidms 570	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥 ⊔ 𝑥) = (𝐴 ⊔ 𝐴))
13	10, 12	breq12d 5043	. . . 4 ⊢ (𝑥 = 𝐴 → (𝑥 ≺ (𝑥 ⊔ 𝑥) ↔ 𝐴 ≺ (𝐴 ⊔ 𝐴)))
14	9, 13	orbi12d 916	. . 3 ⊢ (𝑥 = 𝐴 → ((𝑥 = ∅ ∨ 𝑥 ≺ (𝑥 ⊔ 𝑥)) ↔ (𝐴 = ∅ ∨ 𝐴 ≺ (𝐴 ⊔ 𝐴))))
15	8, 14	elab3 3622	. 2 ⊢ (𝐴 ∈ {𝑥 ∣ (𝑥 = ∅ ∨ 𝑥 ≺ (𝑥 ⊔ 𝑥))} ↔ (𝐴 = ∅ ∨ 𝐴 ≺ (𝐴 ⊔ 𝐴)))
16	2, 15	bitri 278	1 ⊢ (𝐴 ∈ FinV ↔ (𝐴 = ∅ ∨ 𝐴 ≺ (𝐴 ⊔ 𝐴)))

Syntax hints:   ↔ wb 209   ∨ wo 844   = wceq 1538   ∈ wcel 2111  {cab 2776  Vcvv 3441  ∅c0 4243   class class class wbr 5030   ≺ csdm 8491   ⊔ cdju 9311  Fin^Vcfin5 9693

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pr 5295

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-v 3443  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-br 5031  df-opab 5093  df-xp 5525  df-rel 5526  df-dom 8494  df-sdom 8495  df-dju 9314  df-fin5 9700

This theorem is referenced by:  isfin5-2  9802  fin56  9804