Theorem isfin5 10296

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 10286	. . 3 ⊢ FinV = {𝑥 ∣ (𝑥 = ∅ ∨ 𝑥 ≺ (𝑥 ⊔ 𝑥))}
2	1	eleq2i 2825	. 2 ⊢ (𝐴 ∈ FinV ↔ 𝐴 ∈ {𝑥 ∣ (𝑥 = ∅ ∨ 𝑥 ≺ (𝑥 ⊔ 𝑥))})
3		id 22	. . . . 5 ⊢ (𝐴 = ∅ → 𝐴 = ∅)
4		0ex 5307	. . . . 5 ⊢ ∅ ∈ V
5	3, 4	eqeltrdi 2841	. . . 4 ⊢ (𝐴 = ∅ → 𝐴 ∈ V)
6		relsdom 8948	. . . . 5 ⊢ Rel ≺
7	6	brrelex1i 5732	. . . 4 ⊢ (𝐴 ≺ (𝐴 ⊔ 𝐴) → 𝐴 ∈ V)
8	5, 7	jaoi 855	. . 3 ⊢ ((𝐴 = ∅ ∨ 𝐴 ≺ (𝐴 ⊔ 𝐴)) → 𝐴 ∈ V)
9		eqeq1 2736	. . . 4 ⊢ (𝑥 = 𝐴 → (𝑥 = ∅ ↔ 𝐴 = ∅))
10		id 22	. . . . 5 ⊢ (𝑥 = 𝐴 → 𝑥 = 𝐴)
11		djueq12 9901	. . . . . 6 ⊢ ((𝑥 = 𝐴 ∧ 𝑥 = 𝐴) → (𝑥 ⊔ 𝑥) = (𝐴 ⊔ 𝐴))
12	11	anidms 567	. . . . 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 917	. . 3 ⊢ (𝑥 = 𝐴 → ((𝑥 = ∅ ∨ 𝑥 ≺ (𝑥 ⊔ 𝑥)) ↔ (𝐴 = ∅ ∨ 𝐴 ≺ (𝐴 ⊔ 𝐴))))
15	8, 14	elab3 3676	. 2 ⊢ (𝐴 ∈ {𝑥 ∣ (𝑥 = ∅ ∨ 𝑥 ≺ (𝑥 ⊔ 𝑥))} ↔ (𝐴 = ∅ ∨ 𝐴 ≺ (𝐴 ⊔ 𝐴)))
16	2, 15	bitri 274	1 ⊢ (𝐴 ∈ FinV ↔ (𝐴 = ∅ ∨ 𝐴 ≺ (𝐴 ⊔ 𝐴)))

Syntax hints:   ↔ wb 205   ∨ wo 845   = wceq 1541   ∈ wcel 2106  {cab 2709  Vcvv 3474  ∅c0 4322   class class class wbr 5148   ≺ csdm 8940   ⊔ cdju 9895  Fin^Vcfin5 10279

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-br 5149  df-opab 5211  df-xp 5682  df-rel 5683  df-dom 8943  df-sdom 8944  df-dju 9898  df-fin5 10286

This theorem is referenced by:  isfin5-2  10388  fin56  10390