Theorem isfin4 10385

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

Assertion

Ref	Expression
isfin4	⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ FinIV ↔ ¬ ∃𝑦(𝑦 ⊊ 𝐴 ∧ 𝑦 ≈ 𝐴)))

Distinct variable group:   𝑦,𝐴

Allowed substitution hint:   𝑉(𝑦)

Proof of Theorem isfin4

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		psseq2 4020	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑦 ⊊ 𝑥 ↔ 𝑦 ⊊ 𝐴))
2		breq2 5109	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑦 ≈ 𝑥 ↔ 𝑦 ≈ 𝐴))
3	1, 2	anbi12d 644	. . . 4 ⊢ (𝑥 = 𝐴 → ((𝑦 ⊊ 𝑥 ∧ 𝑦 ≈ 𝑥) ↔ (𝑦 ⊊ 𝐴 ∧ 𝑦 ≈ 𝐴)))
4	3	exbidv 1955	. . 3 ⊢ (𝑥 = 𝐴 → (∃𝑦(𝑦 ⊊ 𝑥 ∧ 𝑦 ≈ 𝑥) ↔ ∃𝑦(𝑦 ⊊ 𝐴 ∧ 𝑦 ≈ 𝐴)))
5	4	notbid 321	. 2 ⊢ (𝑥 = 𝐴 → (¬ ∃𝑦(𝑦 ⊊ 𝑥 ∧ 𝑦 ≈ 𝑥) ↔ ¬ ∃𝑦(𝑦 ⊊ 𝐴 ∧ 𝑦 ≈ 𝐴)))
6		df-fin4 10375	. 2 ⊢ FinIV = {𝑥 ∣ ¬ ∃𝑦(𝑦 ⊊ 𝑥 ∧ 𝑦 ≈ 𝑥)}
7	5, 6	elab2g 3636	1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ FinIV ↔ ¬ ∃𝑦(𝑦 ⊊ 𝐴 ∧ 𝑦 ≈ 𝐴)))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1569  ∃wex 1811   ∈ wcel 2146   ⊊ wpss 4015   class class class wbr 5105   ≈ cen 8981  Fin^IVcfin4 10368

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

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 2838  df-ne 2959  df-rab 3416  df-v 3455  df-ss 3900  df-pss 4016  df-dif 4049  df-un 4153  df-nul 4281  df-if 4485  df-sn 4588  df-pr 4590  df-op 4838  df-br 5106  df-fin4 10375

This theorem is used by:  fin4i  10386  fin4en1  10397  ssfin4  10398  infpssALT  10401  isfin4-2  10402