This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem frc

Description: Property of well-founded relation (one direction of definition using class variables). (Contributed by NM, 17-Feb-2004) (Revised by Mario Carneiro, 19-Nov-2014)

Ref Expression
Hypothesis frc.1 ⊢ 𝐵 ∈ V
Assertion frc ( ( 𝑅 Fr 𝐴 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐵 ≠ ∅ ) → ∃ 𝑥 ∈ 𝐵 { 𝑦 ∈ 𝐵 ∣ 𝑦 𝑅 𝑥 } = ∅ )

Proof

Step Hyp Ref Expression
1 frc.1 ⊢ 𝐵 ∈ V
2 fri ⊢ ( ( ( 𝐵 ∈ V ∧ 𝑅 Fr 𝐴 ) ∧ ( 𝐵 ⊆ 𝐴 ∧ 𝐵 ≠ ∅ ) ) → ∃ 𝑥 ∈ 𝐵 ∀ 𝑧 ∈ 𝐵 ¬ 𝑧 𝑅 𝑥 )
3 1 2 mpanl1 ⊢ ( ( 𝑅 Fr 𝐴 ∧ ( 𝐵 ⊆ 𝐴 ∧ 𝐵 ≠ ∅ ) ) → ∃ 𝑥 ∈ 𝐵 ∀ 𝑧 ∈ 𝐵 ¬ 𝑧 𝑅 𝑥 )
4 3 3impb ⊢ ( ( 𝑅 Fr 𝐴 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐵 ≠ ∅ ) → ∃ 𝑥 ∈ 𝐵 ∀ 𝑧 ∈ 𝐵 ¬ 𝑧 𝑅 𝑥 )
5 breq1 ⊢ ( 𝑦 = 𝑧 → ( 𝑦 𝑅 𝑥 ↔ 𝑧 𝑅 𝑥 ) )
6 5 rabeq0w ⊢ ( { 𝑦 ∈ 𝐵 ∣ 𝑦 𝑅 𝑥 } = ∅ ↔ ∀ 𝑧 ∈ 𝐵 ¬ 𝑧 𝑅 𝑥 )
7 6 rexbii ⊢ ( ∃ 𝑥 ∈ 𝐵 { 𝑦 ∈ 𝐵 ∣ 𝑦 𝑅 𝑥 } = ∅ ↔ ∃ 𝑥 ∈ 𝐵 ∀ 𝑧 ∈ 𝐵 ¬ 𝑧 𝑅 𝑥 )
8 4 7 sylibr ⊢ ( ( 𝑅 Fr 𝐴 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐵 ≠ ∅ ) → ∃ 𝑥 ∈ 𝐵 { 𝑦 ∈ 𝐵 ∣ 𝑦 𝑅 𝑥 } = ∅ )