Metamath Proof Explorer


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 → ( 𝐵𝐴 ↔ suc 𝐵𝐴 ) )

Proof

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