Theorem limsuc

Description: The successor of a member of a limit ordinal is also a member. (Contributed by NM, 3-Sep-2003)

	Ref	Expression
Assertion	limsuc	⊢ ( Lim 𝐴 → ( 𝐵 ∈ 𝐴 ↔ suc 𝐵 ∈ 𝐴 ) )

Proof

Step	Hyp	Ref	Expression
1		dflim4	⊢ ( Lim 𝐴 ↔ ( Ord 𝐴 ∧ ∅ ∈ 𝐴 ∧ ∀ 𝑥 ∈ 𝐴 suc 𝑥 ∈ 𝐴 ) )
2		suceq	⊢ ( 𝑥 = 𝐵 → suc 𝑥 = suc 𝐵 )
3	2	eleq1d	⊢ ( 𝑥 = 𝐵 → ( suc 𝑥 ∈ 𝐴 ↔ suc 𝐵 ∈ 𝐴 ) )
4	3	rspccv	⊢ ( ∀ 𝑥 ∈ 𝐴 suc 𝑥 ∈ 𝐴 → ( 𝐵 ∈ 𝐴 → suc 𝐵 ∈ 𝐴 ) )
5	4	3ad2ant3	⊢ ( ( Ord 𝐴 ∧ ∅ ∈ 𝐴 ∧ ∀ 𝑥 ∈ 𝐴 suc 𝑥 ∈ 𝐴 ) → ( 𝐵 ∈ 𝐴 → suc 𝐵 ∈ 𝐴 ) )
6	1 5	sylbi	⊢ ( Lim 𝐴 → ( 𝐵 ∈ 𝐴 → suc 𝐵 ∈ 𝐴 ) )
7		limord	⊢ ( Lim 𝐴 → Ord 𝐴 )
8		ordtr	⊢ ( Ord 𝐴 → Tr 𝐴 )
9		trsuc	⊢ ( ( Tr 𝐴 ∧ suc 𝐵 ∈ 𝐴 ) → 𝐵 ∈ 𝐴 )
10	9	ex	⊢ ( Tr 𝐴 → ( suc 𝐵 ∈ 𝐴 → 𝐵 ∈ 𝐴 ) )
11	7 8 10	3syl	⊢ ( Lim 𝐴 → ( suc 𝐵 ∈ 𝐴 → 𝐵 ∈ 𝐴 ) )
12	6 11	impbid	⊢ ( Lim 𝐴 → ( 𝐵 ∈ 𝐴 ↔ suc 𝐵 ∈ 𝐴 ) )