Theorem isfin4 9713

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 4065	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑦 ⊊ 𝑥 ↔ 𝑦 ⊊ 𝐴))
2		breq2 5063	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑦 ≈ 𝑥 ↔ 𝑦 ≈ 𝐴))
3	1, 2	anbi12d 632	. . . 4 ⊢ (𝑥 = 𝐴 → ((𝑦 ⊊ 𝑥 ∧ 𝑦 ≈ 𝑥) ↔ (𝑦 ⊊ 𝐴 ∧ 𝑦 ≈ 𝐴)))
4	3	exbidv 1918	. . 3 ⊢ (𝑥 = 𝐴 → (∃𝑦(𝑦 ⊊ 𝑥 ∧ 𝑦 ≈ 𝑥) ↔ ∃𝑦(𝑦 ⊊ 𝐴 ∧ 𝑦 ≈ 𝐴)))
5	4	notbid 320	. 2 ⊢ (𝑥 = 𝐴 → (¬ ∃𝑦(𝑦 ⊊ 𝑥 ∧ 𝑦 ≈ 𝑥) ↔ ¬ ∃𝑦(𝑦 ⊊ 𝐴 ∧ 𝑦 ≈ 𝐴)))
6		df-fin4 9703	. 2 ⊢ FinIV = {𝑥 ∣ ¬ ∃𝑦(𝑦 ⊊ 𝑥 ∧ 𝑦 ≈ 𝑥)}
7	5, 6	elab2g 3668	1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ FinIV ↔ ¬ ∃𝑦(𝑦 ⊊ 𝐴 ∧ 𝑦 ≈ 𝐴)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 398   = wceq 1533  ∃wex 1776   ∈ wcel 2110   ⊊ wpss 3937   class class class wbr 5059   ≈ cen 8500  Fin^IVcfin4 9696

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2156  ax-12 2172  ax-ext 2793

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-rab 3147  df-v 3497  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-pss 3954  df-nul 4292  df-if 4468  df-sn 4562  df-pr 4564  df-op 4568  df-br 5060  df-fin4 9703

This theorem is referenced by:  fin4i  9714  fin4en1  9725  ssfin4  9726  infpssALT  9729  isfin4-2  9730