Theorem isfin7 10230

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 3157	. . 3 ⊢ (𝑥 = 𝐴 → (∃𝑦 ∈ (On ∖ ω)𝑥 ≈ 𝑦 ↔ ∃𝑦 ∈ (On ∖ ω)𝐴 ≈ 𝑦))
3	2	notbid 318	. 2 ⊢ (𝑥 = 𝐴 → (¬ ∃𝑦 ∈ (On ∖ ω)𝑥 ≈ 𝑦 ↔ ¬ ∃𝑦 ∈ (On ∖ ω)𝐴 ≈ 𝑦))
4		df-fin7 10220	. 2 ⊢ FinVII = {𝑥 ∣ ¬ ∃𝑦 ∈ (On ∖ ω)𝑥 ≈ 𝑦}
5	3, 4	elab2g 3644	1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ FinVII ↔ ¬ ∃𝑦 ∈ (On ∖ ω)𝐴 ≈ 𝑦))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   = wceq 1540   ∈ wcel 2109  ∃wrex 3053   ∖ cdif 3908   class class class wbr 5102  Oncon0 6320  ωcom 7822   ≈ cen 8892  Fin^VIIcfin7 10213

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-rex 3054  df-rab 3403  df-v 3446  df-dif 3914  df-un 3916  df-ss 3928  df-nul 4293  df-if 4485  df-sn 4586  df-pr 4588  df-op 4592  df-br 5103  df-fin7 10220

This theorem is referenced by:  fin17  10323  fin67  10324  isfin7-2  10325