Theorem isfin5 10337

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 10327	. . 3 ⊢ FinV = {𝑥 ∣ (𝑥 = ∅ ∨ 𝑥 ≺ (𝑥 ⊔ 𝑥))}
2	1	eleq2i 2831	. 2 ⊢ (𝐴 ∈ FinV ↔ 𝐴 ∈ {𝑥 ∣ (𝑥 = ∅ ∨ 𝑥 ≺ (𝑥 ⊔ 𝑥))})
3		id 22	. . . . 5 ⊢ (𝐴 = ∅ → 𝐴 = ∅)
4		0ex 5313	. . . . 5 ⊢ ∅ ∈ V
5	3, 4	eqeltrdi 2847	. . . 4 ⊢ (𝐴 = ∅ → 𝐴 ∈ V)
6		relsdom 8991	. . . . 5 ⊢ Rel ≺
7	6	brrelex1i 5745	. . . 4 ⊢ (𝐴 ≺ (𝐴 ⊔ 𝐴) → 𝐴 ∈ V)
8	5, 7	jaoi 857	. . 3 ⊢ ((𝐴 = ∅ ∨ 𝐴 ≺ (𝐴 ⊔ 𝐴)) → 𝐴 ∈ V)
9		eqeq1 2739	. . . 4 ⊢ (𝑥 = 𝐴 → (𝑥 = ∅ ↔ 𝐴 = ∅))
10		id 22	. . . . 5 ⊢ (𝑥 = 𝐴 → 𝑥 = 𝐴)
11		djueq12 9942	. . . . . 6 ⊢ ((𝑥 = 𝐴 ∧ 𝑥 = 𝐴) → (𝑥 ⊔ 𝑥) = (𝐴 ⊔ 𝐴))
12	11	anidms 566	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥 ⊔ 𝑥) = (𝐴 ⊔ 𝐴))
13	10, 12	breq12d 5161	. . . 4 ⊢ (𝑥 = 𝐴 → (𝑥 ≺ (𝑥 ⊔ 𝑥) ↔ 𝐴 ≺ (𝐴 ⊔ 𝐴)))
14	9, 13	orbi12d 918	. . 3 ⊢ (𝑥 = 𝐴 → ((𝑥 = ∅ ∨ 𝑥 ≺ (𝑥 ⊔ 𝑥)) ↔ (𝐴 = ∅ ∨ 𝐴 ≺ (𝐴 ⊔ 𝐴))))
15	8, 14	elab3 3689	. 2 ⊢ (𝐴 ∈ {𝑥 ∣ (𝑥 = ∅ ∨ 𝑥 ≺ (𝑥 ⊔ 𝑥))} ↔ (𝐴 = ∅ ∨ 𝐴 ≺ (𝐴 ⊔ 𝐴)))
16	2, 15	bitri 275	1 ⊢ (𝐴 ∈ FinV ↔ (𝐴 = ∅ ∨ 𝐴 ≺ (𝐴 ⊔ 𝐴)))

Syntax hints:   ↔ wb 206   ∨ wo 847   = wceq 1537   ∈ wcel 2106  {cab 2712  Vcvv 3478  ∅c0 4339   class class class wbr 5148   ≺ csdm 8983   ⊔ cdju 9936  Fin^Vcfin5 10320

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-opab 5211  df-xp 5695  df-rel 5696  df-dom 8986  df-sdom 8987  df-dju 9939  df-fin5 10327

This theorem is referenced by:  isfin5-2  10429  fin56  10431