Theorem onzsl 7845

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.)

Assertion

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

Distinct variable group:   𝑥,𝐴

Proof of Theorem onzsl

Step	Hyp	Ref	Expression
1		elex 3474	. . 3 ⊢ (𝐴 ∈ On → 𝐴 ∈ V)
2		eloni 6371	. . 3 ⊢ (𝐴 ∈ On → Ord 𝐴)
3		ordzsl 7844	. . . 4 ⊢ (Ord 𝐴 ↔ (𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥 ∨ Lim 𝐴))
4		3mix1 1349	. . . . . 6 ⊢ (𝐴 = ∅ → (𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥 ∨ (𝐴 ∈ V ∧ Lim 𝐴)))
5	4	adantl 487	. . . . 5 ⊢ ((𝐴 ∈ V ∧ 𝐴 = ∅) → (𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥 ∨ (𝐴 ∈ V ∧ Lim 𝐴)))
6		3mix2 1350	. . . . . 6 ⊢ (∃𝑥 ∈ On 𝐴 = suc 𝑥 → (𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥 ∨ (𝐴 ∈ V ∧ Lim 𝐴)))
7	6	adantl 487	. . . . 5 ⊢ ((𝐴 ∈ V ∧ ∃𝑥 ∈ On 𝐴 = suc 𝑥) → (𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥 ∨ (𝐴 ∈ V ∧ Lim 𝐴)))
8		3mix3 1351	. . . . 5 ⊢ ((𝐴 ∈ V ∧ Lim 𝐴) → (𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥 ∨ (𝐴 ∈ V ∧ Lim 𝐴)))
9	5, 7, 8	3jaodan 1458	. . . 4 ⊢ ((𝐴 ∈ V ∧ (𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥 ∨ Lim 𝐴)) → (𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥 ∨ (𝐴 ∈ V ∧ Lim 𝐴)))
10	3, 9	sylan2b 606	. . 3 ⊢ ((𝐴 ∈ V ∧ Ord 𝐴) → (𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥 ∨ (𝐴 ∈ V ∧ Lim 𝐴)))
11	1, 2, 10	syl2anc 596	. 2 ⊢ (𝐴 ∈ On → (𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥 ∨ (𝐴 ∈ V ∧ Lim 𝐴)))
12		0elon 6417	. . . 4 ⊢ ∅ ∈ On
13		eleq1 2850	. . . 4 ⊢ (𝐴 = ∅ → (𝐴 ∈ On ↔ ∅ ∈ On))
14	12, 13	mpbiri 261	. . 3 ⊢ (𝐴 = ∅ → 𝐴 ∈ On)
15		onsuc 7812	. . . . 5 ⊢ (𝑥 ∈ On → suc 𝑥 ∈ On)
16		eleq1 2850	. . . . 5 ⊢ (𝐴 = suc 𝑥 → (𝐴 ∈ On ↔ suc 𝑥 ∈ On))
17	15, 16	syl5ibrcom 250	. . . 4 ⊢ (𝑥 ∈ On → (𝐴 = suc 𝑥 → 𝐴 ∈ On))
18	17	rexlimiv 3158	. . 3 ⊢ (∃𝑥 ∈ On 𝐴 = suc 𝑥 → 𝐴 ∈ On)
19		limelon 6427	. . 3 ⊢ ((𝐴 ∈ V ∧ Lim 𝐴) → 𝐴 ∈ On)
20	14, 18, 19	3jaoi 1454	. 2 ⊢ ((𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥 ∨ (𝐴 ∈ V ∧ Lim 𝐴)) → 𝐴 ∈ On)
21	11, 20	impbii 212	1 ⊢ (𝐴 ∈ On ↔ (𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥 ∨ (𝐴 ∈ V ∧ Lim 𝐴)))

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 401   ∨ w3o 1102   = wceq 1570   ∈ wcel 2145  ∃wrex 3088  Vcvv 3453  ∅c0 4282  Ord word 6360  Oncon0 6361  Lim wlim 6362  suc csuc 6363

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2734  ax-sep 5255  ax-nul 5267  ax-pr 5402  ax-un 7739

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2741  df-cleq 2754  df-clel 2837  df-ne 2958  df-ral 3079  df-rex 3089  df-rab 3415  df-v 3455  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-br 5108  df-opab 5172  df-tr 5217  df-eprel 5559  df-po 5567  df-so 5568  df-fr 5612  df-we 5614  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367

This theorem is used by:  oawordeulem  8544  r1pwss  9769  r1val1  9771  pwcfsdom  10595  winalim2  10708  rankcf  10789  dfrdg4  36517  naddwordnexlem4  44229