Theorem onnbtwn

Description: There is no set between an ordinal number and its successor. Proposition 7.25 of TakeutiZaring p. 41. (Contributed by NM, 9-Jun-1994)

	Ref	Expression
Assertion	onnbtwn	⊢ ( 𝐴 ∈ On → ¬ ( 𝐴 ∈ 𝐵 ∧ 𝐵 ∈ suc 𝐴 ) )

Proof

Step	Hyp	Ref	Expression
1		eloni	⊢ ( 𝐴 ∈ On → Ord 𝐴 )
2		ordnbtwn	⊢ ( Ord 𝐴 → ¬ ( 𝐴 ∈ 𝐵 ∧ 𝐵 ∈ suc 𝐴 ) )
3	1 2	syl	⊢ ( 𝐴 ∈ On → ¬ ( 𝐴 ∈ 𝐵 ∧ 𝐵 ∈ suc 𝐴 ) )