Theorem efrn2lp

Description: A well-founded class contains no 2-cycle loops. (Contributed by NM, 19-Apr-1994)

	Ref	Expression
Assertion	efrn2lp	⊢ ( ( E Fr 𝐴 ∧ ( 𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴 ) ) → ¬ ( 𝐵 ∈ 𝐶 ∧ 𝐶 ∈ 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		fr2nr	⊢ ( ( E Fr 𝐴 ∧ ( 𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴 ) ) → ¬ ( 𝐵 E 𝐶 ∧ 𝐶 E 𝐵 ) )
2		epelg	⊢ ( 𝐶 ∈ 𝐴 → ( 𝐵 E 𝐶 ↔ 𝐵 ∈ 𝐶 ) )
3		epelg	⊢ ( 𝐵 ∈ 𝐴 → ( 𝐶 E 𝐵 ↔ 𝐶 ∈ 𝐵 ) )
4	2 3	bi2anan9r	⊢ ( ( 𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴 ) → ( ( 𝐵 E 𝐶 ∧ 𝐶 E 𝐵 ) ↔ ( 𝐵 ∈ 𝐶 ∧ 𝐶 ∈ 𝐵 ) ) )
5	4	adantl	⊢ ( ( E Fr 𝐴 ∧ ( 𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴 ) ) → ( ( 𝐵 E 𝐶 ∧ 𝐶 E 𝐵 ) ↔ ( 𝐵 ∈ 𝐶 ∧ 𝐶 ∈ 𝐵 ) ) )
6	1 5	mtbid	⊢ ( ( E Fr 𝐴 ∧ ( 𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴 ) ) → ¬ ( 𝐵 ∈ 𝐶 ∧ 𝐶 ∈ 𝐵 ) )