Theorem nordeq

Description: A member of an ordinal class is not equal to it. (Contributed by NM, 25-May-1998)

	Ref	Expression
Assertion	nordeq	⊢ ( ( Ord 𝐴 ∧ 𝐵 ∈ 𝐴 ) → 𝐴 ≠ 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		ordirr	⊢ ( Ord 𝐴 → ¬ 𝐴 ∈ 𝐴 )
2		eleq1	⊢ ( 𝐴 = 𝐵 → ( 𝐴 ∈ 𝐴 ↔ 𝐵 ∈ 𝐴 ) )
3	2	notbid	⊢ ( 𝐴 = 𝐵 → ( ¬ 𝐴 ∈ 𝐴 ↔ ¬ 𝐵 ∈ 𝐴 ) )
4	1 3	syl5ibcom	⊢ ( Ord 𝐴 → ( 𝐴 = 𝐵 → ¬ 𝐵 ∈ 𝐴 ) )
5	4	necon2ad	⊢ ( Ord 𝐴 → ( 𝐵 ∈ 𝐴 → 𝐴 ≠ 𝐵 ) )
6	5	imp	⊢ ( ( Ord 𝐴 ∧ 𝐵 ∈ 𝐴 ) → 𝐴 ≠ 𝐵 )