Metamath Proof Explorer


Theorem isfin7

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

Ref Expression
Assertion isfin7 ( 𝐴𝑉 → ( 𝐴 ∈ FinVII ↔ ¬ ∃ 𝑦 ∈ ( On ∖ ω ) 𝐴𝑦 ) )

Proof

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