Theorem dflim7

Description: A limit ordinal is a nonzero ordinal that contains all the successors of its elements. Lemma 1.18 of Schloeder p. 2. Closely related to dflim4 . (Contributed by RP, 17-Jan-2025)

	Ref	Expression
Assertion	dflim7	⊢ ( Lim 𝐴 ↔ ( Ord 𝐴 ∧ ∀ 𝑏 ∈ 𝐴 suc 𝑏 ∈ 𝐴 ∧ 𝐴 ≠ ∅ ) )

Proof

Step	Hyp	Ref	Expression
1		dflim4	⊢ ( Lim 𝐴 ↔ ( Ord 𝐴 ∧ ∅ ∈ 𝐴 ∧ ∀ 𝑏 ∈ 𝐴 suc 𝑏 ∈ 𝐴 ) )
2		ord0eln0	⊢ ( Ord 𝐴 → ( ∅ ∈ 𝐴 ↔ 𝐴 ≠ ∅ ) )
3	2	anbi1d	⊢ ( Ord 𝐴 → ( ( ∅ ∈ 𝐴 ∧ ∀ 𝑏 ∈ 𝐴 suc 𝑏 ∈ 𝐴 ) ↔ ( 𝐴 ≠ ∅ ∧ ∀ 𝑏 ∈ 𝐴 suc 𝑏 ∈ 𝐴 ) ) )
4	3	biancomd	⊢ ( Ord 𝐴 → ( ( ∅ ∈ 𝐴 ∧ ∀ 𝑏 ∈ 𝐴 suc 𝑏 ∈ 𝐴 ) ↔ ( ∀ 𝑏 ∈ 𝐴 suc 𝑏 ∈ 𝐴 ∧ 𝐴 ≠ ∅ ) ) )
5	4	pm5.32i	⊢ ( ( Ord 𝐴 ∧ ( ∅ ∈ 𝐴 ∧ ∀ 𝑏 ∈ 𝐴 suc 𝑏 ∈ 𝐴 ) ) ↔ ( Ord 𝐴 ∧ ( ∀ 𝑏 ∈ 𝐴 suc 𝑏 ∈ 𝐴 ∧ 𝐴 ≠ ∅ ) ) )
6		3anass	⊢ ( ( Ord 𝐴 ∧ ∅ ∈ 𝐴 ∧ ∀ 𝑏 ∈ 𝐴 suc 𝑏 ∈ 𝐴 ) ↔ ( Ord 𝐴 ∧ ( ∅ ∈ 𝐴 ∧ ∀ 𝑏 ∈ 𝐴 suc 𝑏 ∈ 𝐴 ) ) )
7		3anass	⊢ ( ( Ord 𝐴 ∧ ∀ 𝑏 ∈ 𝐴 suc 𝑏 ∈ 𝐴 ∧ 𝐴 ≠ ∅ ) ↔ ( Ord 𝐴 ∧ ( ∀ 𝑏 ∈ 𝐴 suc 𝑏 ∈ 𝐴 ∧ 𝐴 ≠ ∅ ) ) )
8	5 6 7	3bitr4i	⊢ ( ( Ord 𝐴 ∧ ∅ ∈ 𝐴 ∧ ∀ 𝑏 ∈ 𝐴 suc 𝑏 ∈ 𝐴 ) ↔ ( Ord 𝐴 ∧ ∀ 𝑏 ∈ 𝐴 suc 𝑏 ∈ 𝐴 ∧ 𝐴 ≠ ∅ ) )
9	1 8	bitri	⊢ ( Lim 𝐴 ↔ ( Ord 𝐴 ∧ ∀ 𝑏 ∈ 𝐴 suc 𝑏 ∈ 𝐴 ∧ 𝐴 ≠ ∅ ) )