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Metamath Proof Explorer

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 onsuc ⊢ ( 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 ) ) )