Theorem ordelssne

Description: For ordinal classes, membership is equivalent to strict inclusion. Corollary 7.8 of TakeutiZaring p. 37. (Contributed by NM, 25-Nov-1995)

	Ref	Expression
Assertion	ordelssne	⊢ ( ( Ord 𝐴 ∧ Ord 𝐵 ) → ( 𝐴 ∈ 𝐵 ↔ ( 𝐴 ⊆ 𝐵 ∧ 𝐴 ≠ 𝐵 ) ) )

Proof

Step	Hyp	Ref	Expression
1		ordtr	⊢ ( Ord 𝐴 → Tr 𝐴 )
2		tz7.7	⊢ ( ( Ord 𝐵 ∧ Tr 𝐴 ) → ( 𝐴 ∈ 𝐵 ↔ ( 𝐴 ⊆ 𝐵 ∧ 𝐴 ≠ 𝐵 ) ) )
3	1 2	sylan2	⊢ ( ( Ord 𝐵 ∧ Ord 𝐴 ) → ( 𝐴 ∈ 𝐵 ↔ ( 𝐴 ⊆ 𝐵 ∧ 𝐴 ≠ 𝐵 ) ) )
4	3	ancoms	⊢ ( ( Ord 𝐴 ∧ Ord 𝐵 ) → ( 𝐴 ∈ 𝐵 ↔ ( 𝐴 ⊆ 𝐵 ∧ 𝐴 ≠ 𝐵 ) ) )