Theorem isfin7 10260

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

Assertion

Ref	Expression
isfin7	⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ FinVII ↔ ¬ ∃𝑦 ∈ (On ∖ ω)𝐴 ≈ 𝑦))

Distinct variable group:   𝑦,𝐴

Allowed substitution hint:   𝑉(𝑦)

Proof of Theorem isfin7

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		breq1 5105	. . . 4 ⊢ (𝑥 = 𝐴 → (𝑥 ≈ 𝑦 ↔ 𝐴 ≈ 𝑦))
2	1	rexbidv 3188	. . 3 ⊢ (𝑥 = 𝐴 → (∃𝑦 ∈ (On ∖ ω)𝑥 ≈ 𝑦 ↔ ∃𝑦 ∈ (On ∖ ω)𝐴 ≈ 𝑦))
3	2	notbid 320	. 2 ⊢ (𝑥 = 𝐴 → (¬ ∃𝑦 ∈ (On ∖ ω)𝑥 ≈ 𝑦 ↔ ¬ ∃𝑦 ∈ (On ∖ ω)𝐴 ≈ 𝑦))
4		df-fin7 10250	. 2 ⊢ FinVII = {𝑥 ∣ ¬ ∃𝑦 ∈ (On ∖ ω)𝑥 ≈ 𝑦}
5	3, 4	elab2g 3641	1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ FinVII ↔ ¬ ∃𝑦 ∈ (On ∖ ω)𝐴 ≈ 𝑦))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   = wceq 1562   ∈ wcel 2144  ∃wrex 3088   ∖ cdif 3903   class class class wbr 5102  Oncon0 6348  ωcom 7848   ≈ cen 8926  Fin^VIIcfin7 10243

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-ext 2736

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1101  df-tru 1565  df-fal 1575  df-ex 1802  df-sb 2093  df-clab 2743  df-cleq 2756  df-clel 2839  df-rex 3089  df-rab 3417  df-v 3458  df-dif 3909  df-un 3911  df-ss 3923  df-nul 4288  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-br 5103  df-fin7 10250

This theorem is referenced by:  fin17  10353  fin67  10354  isfin7-2  10355