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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 → ( 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 ) )