Theorem isfin7

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

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

Proof

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