Theorem isfin4 10366

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 4114	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑦 ⊊ 𝑥 ↔ 𝑦 ⊊ 𝐴))
2		breq2 5170	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑦 ≈ 𝑥 ↔ 𝑦 ≈ 𝐴))
3	1, 2	anbi12d 631	. . . 4 ⊢ (𝑥 = 𝐴 → ((𝑦 ⊊ 𝑥 ∧ 𝑦 ≈ 𝑥) ↔ (𝑦 ⊊ 𝐴 ∧ 𝑦 ≈ 𝐴)))
4	3	exbidv 1920	. . 3 ⊢ (𝑥 = 𝐴 → (∃𝑦(𝑦 ⊊ 𝑥 ∧ 𝑦 ≈ 𝑥) ↔ ∃𝑦(𝑦 ⊊ 𝐴 ∧ 𝑦 ≈ 𝐴)))
5	4	notbid 318	. 2 ⊢ (𝑥 = 𝐴 → (¬ ∃𝑦(𝑦 ⊊ 𝑥 ∧ 𝑦 ≈ 𝑥) ↔ ¬ ∃𝑦(𝑦 ⊊ 𝐴 ∧ 𝑦 ≈ 𝐴)))
6		df-fin4 10356	. 2 ⊢ FinIV = {𝑥 ∣ ¬ ∃𝑦(𝑦 ⊊ 𝑥 ∧ 𝑦 ≈ 𝑥)}
7	5, 6	elab2g 3696	1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ FinIV ↔ ¬ ∃𝑦(𝑦 ⊊ 𝐴 ∧ 𝑦 ≈ 𝐴)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1537  ∃wex 1777   ∈ wcel 2108   ⊊ wpss 3977   class class class wbr 5166   ≈ cen 9000  Fin^IVcfin4 10349

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-ne 2947  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-br 5167  df-fin4 10356

This theorem is referenced by:  fin4i  10367  fin4en1  10378  ssfin4  10379  infpssALT  10382  isfin4-2  10383