Theorem isfin7 10392

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 5109	. . . 4 ⊢ (𝑥 = 𝐴 → (𝑥 ≈ 𝑦 ↔ 𝐴 ≈ 𝑦))
2	1	rexbidv 3190	. . 3 ⊢ (𝑥 = 𝐴 → (∃𝑦 ∈ (On ∖ ω)𝑥 ≈ 𝑦 ↔ ∃𝑦 ∈ (On ∖ ω)𝐴 ≈ 𝑦))
3	2	notbid 321	. 2 ⊢ (𝑥 = 𝐴 → (¬ ∃𝑦 ∈ (On ∖ ω)𝑥 ≈ 𝑦 ↔ ¬ ∃𝑦 ∈ (On ∖ ω)𝐴 ≈ 𝑦))
4		df-fin7 10382	. 2 ⊢ FinVII = {𝑥 ∣ ¬ ∃𝑦 ∈ (On ∖ ω)𝑥 ≈ 𝑦}
5	3, 4	elab2g 3637	1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ FinVII ↔ ¬ ∃𝑦 ∈ (On ∖ ω)𝐴 ≈ 𝑦))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   = wceq 1569   ∈ wcel 2146  ∃wrex 3090   ∖ cdif 4048   class class class wbr 5106  Oncon0 6369  ωcom 7887   ≈ cen 8984  Fin^VIIcfin7 10375

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 2839  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-br 5107  df-fin7 10382

This theorem is used by:  fin17  10485  fin67  10486  isfin7-2  10487