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 } = ∅ )