Theorem isfin5 9121

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 9111	. . 3 ⊢ FinV = {𝑥 ∣ (𝑥 = ∅ ∨ 𝑥 ≺ (𝑥 +𝑐 𝑥))}
2	1	eleq2i 2693	. 2 ⊢ (𝐴 ∈ FinV ↔ 𝐴 ∈ {𝑥 ∣ (𝑥 = ∅ ∨ 𝑥 ≺ (𝑥 +𝑐 𝑥))})
3		id 22	. . . . 5 ⊢ (𝐴 = ∅ → 𝐴 = ∅)
4		0ex 4790	. . . . 5 ⊢ ∅ ∈ V
5	3, 4	syl6eqel 2709	. . . 4 ⊢ (𝐴 = ∅ → 𝐴 ∈ V)
6		relsdom 7962	. . . . 5 ⊢ Rel ≺
7	6	brrelexi 5158	. . . 4 ⊢ (𝐴 ≺ (𝐴 +𝑐 𝐴) → 𝐴 ∈ V)
8	5, 7	jaoi 394	. . 3 ⊢ ((𝐴 = ∅ ∨ 𝐴 ≺ (𝐴 +𝑐 𝐴)) → 𝐴 ∈ V)
9		eqeq1 2626	. . . 4 ⊢ (𝑥 = 𝐴 → (𝑥 = ∅ ↔ 𝐴 = ∅))
10		id 22	. . . . 5 ⊢ (𝑥 = 𝐴 → 𝑥 = 𝐴)
11	10, 10	oveq12d 6668	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥 +𝑐 𝑥) = (𝐴 +𝑐 𝐴))
12	10, 11	breq12d 4666	. . . 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 3358	. 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 383   = wceq 1483   ∈ wcel 1990  {cab 2608  Vcvv 3200  ∅c0 3915   class class class wbr 4653  (class class class)co 6650   ≺ csdm 7954   +_𝑐 ccda 8989  Fin^Vcfin5 9104

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-xp 5120  df-rel 5121  df-iota 5851  df-fv 5896  df-ov 6653  df-dom 7957  df-sdom 7958  df-fin5 9111

This theorem is referenced by:  isfin5-2  9213  fin56  9215