Theorem isfin4 10296

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 4046	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑦 ⊊ 𝑥 ↔ 𝑦 ⊊ 𝐴))
2		breq2 5115	. . . . 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 1954	. . 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 10286	. 2 ⊢ FinIV = {𝑥 ∣ ¬ ∃𝑦(𝑦 ⊊ 𝑥 ∧ 𝑦 ≈ 𝑥)}
7	5, 6	elab2g 3641	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 1570  ∃wex 1812   ∈ wcel 2146   ⊊ wpss 3907   class class class wbr 5111   ≈ cen 8946  Fin^IVcfin4 10279

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2737

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2744  df-cleq 2757  df-clel 2840  df-ne 2961  df-rab 3419  df-v 3459  df-dif 3909  df-un 3911  df-ss 3923  df-pss 3926  df-nul 4287  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-br 5112  df-fin4 10286

This theorem is used by:  fin4i  10297  fin4en1  10308  ssfin4  10309  infpssALT  10312  isfin4-2  10313