Theorem isfin5 10390

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 10380	. . 3 ⊢ FinV = {𝑥 ∣ (𝑥 = ∅ ∨ 𝑥 ≺ (𝑥 ⊔ 𝑥))}
2	1	eleq2i 2856	. 2 ⊢ (𝐴 ∈ FinV ↔ 𝐴 ∈ {𝑥 ∣ (𝑥 = ∅ ∨ 𝑥 ≺ (𝑥 ⊔ 𝑥))})
3		id 23	. . . . 5 ⊢ (𝐴 = ∅ → 𝐴 = ∅)
4		0ex 5264	. . . . 5 ⊢ ∅ ∈ V
5	3, 4	eqeltrdi 2872	. . . 4 ⊢ (𝐴 = ∅ → 𝐴 ∈ V)
6		relsdom 8994	. . . . 5 ⊢ Rel ≺
7	6	brrelex1i 5711	. . . 4 ⊢ (𝐴 ≺ (𝐴 ⊔ 𝐴) → 𝐴 ∈ V)
8	5, 7	jaoi 871	. . 3 ⊢ ((𝐴 = ∅ ∨ 𝐴 ≺ (𝐴 ⊔ 𝐴)) → 𝐴 ∈ V)
9		eqeq1 2767	. . . 4 ⊢ (𝑥 = 𝐴 → (𝑥 = ∅ ↔ 𝐴 = ∅))
10		id 23	. . . . 5 ⊢ (𝑥 = 𝐴 → 𝑥 = 𝐴)
11		djueq12 9998	. . . . . 6 ⊢ ((𝑥 = 𝐴 ∧ 𝑥 = 𝐴) → (𝑥 ⊔ 𝑥) = (𝐴 ⊔ 𝐴))
12	11	anidms 577	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥 ⊔ 𝑥) = (𝐴 ⊔ 𝐴))
13	10, 12	breq12d 5119	. . . 4 ⊢ (𝑥 = 𝐴 → (𝑥 ≺ (𝑥 ⊔ 𝑥) ↔ 𝐴 ≺ (𝐴 ⊔ 𝐴)))
14	9, 13	orbi12d 932	. . 3 ⊢ (𝑥 = 𝐴 → ((𝑥 = ∅ ∨ 𝑥 ≺ (𝑥 ⊔ 𝑥)) ↔ (𝐴 = ∅ ∨ 𝐴 ≺ (𝐴 ⊔ 𝐴))))
15	8, 14	elab3 3643	. 2 ⊢ (𝐴 ∈ {𝑥 ∣ (𝑥 = ∅ ∨ 𝑥 ≺ (𝑥 ⊔ 𝑥))} ↔ (𝐴 = ∅ ∨ 𝐴 ≺ (𝐴 ⊔ 𝐴)))
16	2, 15	bitri 278	1 ⊢ (𝐴 ∈ FinV ↔ (𝐴 = ∅ ∨ 𝐴 ≺ (𝐴 ⊔ 𝐴)))

This proof depends on syntax axioms:   ↔ wb 209   ∨ wo 861   = wceq 1569   ∈ wcel 2146  {cab 2741  Vcvv 3454  ∅c0 4281   class class class wbr 5106   ≺ csdm 8986   ⊔ cdju 9992  Fin^Vcfin5 10373

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-9 2156  ax-ext 2735  ax-sep 5252  ax-nul 5263  ax-pr 5395

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2101  df-clab 2742  df-cleq 2755  df-clel 2839  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-br 5107  df-opab 5171  df-xp 5661  df-rel 5662  df-dom 8989  df-sdom 8990  df-dju 9995  df-fin5 10380

This theorem is used by:  isfin5-2  10482  fin56  10484