Theorem dflim2

Description: An alternate definition of a limit ordinal. (Contributed by NM, 4-Nov-2004)

	Ref	Expression
Assertion	dflim2	⊢ ( Lim 𝐴 ↔ ( Ord 𝐴 ∧ ∅ ∈ 𝐴 ∧ 𝐴 = ∪ 𝐴 ) )

Proof

Step	Hyp	Ref	Expression
1		df-lim	⊢ ( Lim 𝐴 ↔ ( Ord 𝐴 ∧ 𝐴 ≠ ∅ ∧ 𝐴 = ∪ 𝐴 ) )
2		ord0eln0	⊢ ( Ord 𝐴 → ( ∅ ∈ 𝐴 ↔ 𝐴 ≠ ∅ ) )
3	2	anbi1d	⊢ ( Ord 𝐴 → ( ( ∅ ∈ 𝐴 ∧ 𝐴 = ∪ 𝐴 ) ↔ ( 𝐴 ≠ ∅ ∧ 𝐴 = ∪ 𝐴 ) ) )
4	3	pm5.32i	⊢ ( ( Ord 𝐴 ∧ ( ∅ ∈ 𝐴 ∧ 𝐴 = ∪ 𝐴 ) ) ↔ ( Ord 𝐴 ∧ ( 𝐴 ≠ ∅ ∧ 𝐴 = ∪ 𝐴 ) ) )
5		3anass	⊢ ( ( Ord 𝐴 ∧ ∅ ∈ 𝐴 ∧ 𝐴 = ∪ 𝐴 ) ↔ ( Ord 𝐴 ∧ ( ∅ ∈ 𝐴 ∧ 𝐴 = ∪ 𝐴 ) ) )
6		3anass	⊢ ( ( Ord 𝐴 ∧ 𝐴 ≠ ∅ ∧ 𝐴 = ∪ 𝐴 ) ↔ ( Ord 𝐴 ∧ ( 𝐴 ≠ ∅ ∧ 𝐴 = ∪ 𝐴 ) ) )
7	4 5 6	3bitr4i	⊢ ( ( Ord 𝐴 ∧ ∅ ∈ 𝐴 ∧ 𝐴 = ∪ 𝐴 ) ↔ ( Ord 𝐴 ∧ 𝐴 ≠ ∅ ∧ 𝐴 = ∪ 𝐴 ) )
8	1 7	bitr4i	⊢ ( Lim 𝐴 ↔ ( Ord 𝐴 ∧ ∅ ∈ 𝐴 ∧ 𝐴 = ∪ 𝐴 ) )