Theorem onzsl

Description: An ordinal number is zero, a successor ordinal, or a limit ordinal number. (Contributed by NM, 1-Oct-2003) (Proof shortened by Andrew Salmon, 27-Aug-2011)

	Ref	Expression
Assertion	onzsl	⊢ ( 𝐴 ∈ On ↔ ( 𝐴 = ∅ ∨ ∃ 𝑥 ∈ On 𝐴 = suc 𝑥 ∨ ( 𝐴 ∈ V ∧ Lim 𝐴 ) ) )

Proof

Step	Hyp	Ref	Expression
1		elex	⊢ ( 𝐴 ∈ On → 𝐴 ∈ V )
2		eloni	⊢ ( 𝐴 ∈ On → Ord 𝐴 )
3		ordzsl	⊢ ( Ord 𝐴 ↔ ( 𝐴 = ∅ ∨ ∃ 𝑥 ∈ On 𝐴 = suc 𝑥 ∨ Lim 𝐴 ) )
4		3mix1	⊢ ( 𝐴 = ∅ → ( 𝐴 = ∅ ∨ ∃ 𝑥 ∈ On 𝐴 = suc 𝑥 ∨ ( 𝐴 ∈ V ∧ Lim 𝐴 ) ) )
5	4	adantl	⊢ ( ( 𝐴 ∈ V ∧ 𝐴 = ∅ ) → ( 𝐴 = ∅ ∨ ∃ 𝑥 ∈ On 𝐴 = suc 𝑥 ∨ ( 𝐴 ∈ V ∧ Lim 𝐴 ) ) )
6		3mix2	⊢ ( ∃ 𝑥 ∈ On 𝐴 = suc 𝑥 → ( 𝐴 = ∅ ∨ ∃ 𝑥 ∈ On 𝐴 = suc 𝑥 ∨ ( 𝐴 ∈ V ∧ Lim 𝐴 ) ) )
7	6	adantl	⊢ ( ( 𝐴 ∈ V ∧ ∃ 𝑥 ∈ On 𝐴 = suc 𝑥 ) → ( 𝐴 = ∅ ∨ ∃ 𝑥 ∈ On 𝐴 = suc 𝑥 ∨ ( 𝐴 ∈ V ∧ Lim 𝐴 ) ) )
8		3mix3	⊢ ( ( 𝐴 ∈ V ∧ Lim 𝐴 ) → ( 𝐴 = ∅ ∨ ∃ 𝑥 ∈ On 𝐴 = suc 𝑥 ∨ ( 𝐴 ∈ V ∧ Lim 𝐴 ) ) )
9	5 7 8	3jaodan	⊢ ( ( 𝐴 ∈ V ∧ ( 𝐴 = ∅ ∨ ∃ 𝑥 ∈ On 𝐴 = suc 𝑥 ∨ Lim 𝐴 ) ) → ( 𝐴 = ∅ ∨ ∃ 𝑥 ∈ On 𝐴 = suc 𝑥 ∨ ( 𝐴 ∈ V ∧ Lim 𝐴 ) ) )
10	3 9	sylan2b	⊢ ( ( 𝐴 ∈ V ∧ Ord 𝐴 ) → ( 𝐴 = ∅ ∨ ∃ 𝑥 ∈ On 𝐴 = suc 𝑥 ∨ ( 𝐴 ∈ V ∧ Lim 𝐴 ) ) )
11	1 2 10	syl2anc	⊢ ( 𝐴 ∈ On → ( 𝐴 = ∅ ∨ ∃ 𝑥 ∈ On 𝐴 = suc 𝑥 ∨ ( 𝐴 ∈ V ∧ Lim 𝐴 ) ) )
12		0elon	⊢ ∅ ∈ On
13		eleq1	⊢ ( 𝐴 = ∅ → ( 𝐴 ∈ On ↔ ∅ ∈ On ) )
14	12 13	mpbiri	⊢ ( 𝐴 = ∅ → 𝐴 ∈ On )
15		suceloni	⊢ ( 𝑥 ∈ On → suc 𝑥 ∈ On )
16		eleq1	⊢ ( 𝐴 = suc 𝑥 → ( 𝐴 ∈ On ↔ suc 𝑥 ∈ On ) )
17	15 16	syl5ibrcom	⊢ ( 𝑥 ∈ On → ( 𝐴 = suc 𝑥 → 𝐴 ∈ On ) )
18	17	rexlimiv	⊢ ( ∃ 𝑥 ∈ On 𝐴 = suc 𝑥 → 𝐴 ∈ On )
19		limelon	⊢ ( ( 𝐴 ∈ V ∧ Lim 𝐴 ) → 𝐴 ∈ On )
20	14 18 19	3jaoi	⊢ ( ( 𝐴 = ∅ ∨ ∃ 𝑥 ∈ On 𝐴 = suc 𝑥 ∨ ( 𝐴 ∈ V ∧ Lim 𝐴 ) ) → 𝐴 ∈ On )
21	11 20	impbii	⊢ ( 𝐴 ∈ On ↔ ( 𝐴 = ∅ ∨ ∃ 𝑥 ∈ On 𝐴 = suc 𝑥 ∨ ( 𝐴 ∈ V ∧ Lim 𝐴 ) ) )