Theorem dflim4

Description: An alternate definition of a limit ordinal. (Contributed by NM, 1-Feb-2005)

	Ref	Expression
Assertion	dflim4	⊢ ( Lim 𝐴 ↔ ( Ord 𝐴 ∧ ∅ ∈ 𝐴 ∧ ∀ 𝑥 ∈ 𝐴 suc 𝑥 ∈ 𝐴 ) )

Proof

Step	Hyp	Ref	Expression
1		dflim2	⊢ ( Lim 𝐴 ↔ ( Ord 𝐴 ∧ ∅ ∈ 𝐴 ∧ 𝐴 = ∪ 𝐴 ) )
2		ordunisuc2	⊢ ( Ord 𝐴 → ( 𝐴 = ∪ 𝐴 ↔ ∀ 𝑥 ∈ 𝐴 suc 𝑥 ∈ 𝐴 ) )
3	2	anbi2d	⊢ ( Ord 𝐴 → ( ( ∅ ∈ 𝐴 ∧ 𝐴 = ∪ 𝐴 ) ↔ ( ∅ ∈ 𝐴 ∧ ∀ 𝑥 ∈ 𝐴 suc 𝑥 ∈ 𝐴 ) ) )
4	3	pm5.32i	⊢ ( ( Ord 𝐴 ∧ ( ∅ ∈ 𝐴 ∧ 𝐴 = ∪ 𝐴 ) ) ↔ ( Ord 𝐴 ∧ ( ∅ ∈ 𝐴 ∧ ∀ 𝑥 ∈ 𝐴 suc 𝑥 ∈ 𝐴 ) ) )
5		3anass	⊢ ( ( Ord 𝐴 ∧ ∅ ∈ 𝐴 ∧ 𝐴 = ∪ 𝐴 ) ↔ ( Ord 𝐴 ∧ ( ∅ ∈ 𝐴 ∧ 𝐴 = ∪ 𝐴 ) ) )
6		3anass	⊢ ( ( Ord 𝐴 ∧ ∅ ∈ 𝐴 ∧ ∀ 𝑥 ∈ 𝐴 suc 𝑥 ∈ 𝐴 ) ↔ ( Ord 𝐴 ∧ ( ∅ ∈ 𝐴 ∧ ∀ 𝑥 ∈ 𝐴 suc 𝑥 ∈ 𝐴 ) ) )
7	4 5 6	3bitr4i	⊢ ( ( Ord 𝐴 ∧ ∅ ∈ 𝐴 ∧ 𝐴 = ∪ 𝐴 ) ↔ ( Ord 𝐴 ∧ ∅ ∈ 𝐴 ∧ ∀ 𝑥 ∈ 𝐴 suc 𝑥 ∈ 𝐴 ) )
8	1 7	bitri	⊢ ( Lim 𝐴 ↔ ( Ord 𝐴 ∧ ∅ ∈ 𝐴 ∧ ∀ 𝑥 ∈ 𝐴 suc 𝑥 ∈ 𝐴 ) )