Theorem onzsl 7825

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 3471	. . 3 ⊢ (𝐴 ∈ On → 𝐴 ∈ V)
2		eloni 6345	. . 3 ⊢ (𝐴 ∈ On → Ord 𝐴)
3		ordzsl 7824	. . . 4 ⊢ (Ord 𝐴 ↔ (𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥 ∨ Lim 𝐴))
4		3mix1 1331	. . . . . 6 ⊢ (𝐴 = ∅ → (𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥 ∨ (𝐴 ∈ V ∧ Lim 𝐴)))
5	4	adantl 481	. . . . 5 ⊢ ((𝐴 ∈ V ∧ 𝐴 = ∅) → (𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥 ∨ (𝐴 ∈ V ∧ Lim 𝐴)))
6		3mix2 1332	. . . . . 6 ⊢ (∃𝑥 ∈ On 𝐴 = suc 𝑥 → (𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥 ∨ (𝐴 ∈ V ∧ Lim 𝐴)))
7	6	adantl 481	. . . . 5 ⊢ ((𝐴 ∈ V ∧ ∃𝑥 ∈ On 𝐴 = suc 𝑥) → (𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥 ∨ (𝐴 ∈ V ∧ Lim 𝐴)))
8		3mix3 1333	. . . . 5 ⊢ ((𝐴 ∈ V ∧ Lim 𝐴) → (𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥 ∨ (𝐴 ∈ V ∧ Lim 𝐴)))
9	5, 7, 8	3jaodan 1433	. . . 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 594	. . 3 ⊢ ((𝐴 ∈ V ∧ Ord 𝐴) → (𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥 ∨ (𝐴 ∈ V ∧ Lim 𝐴)))
11	1, 2, 10	syl2anc 584	. 2 ⊢ (𝐴 ∈ On → (𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥 ∨ (𝐴 ∈ V ∧ Lim 𝐴)))
12		0elon 6390	. . . 4 ⊢ ∅ ∈ On
13		eleq1 2817	. . . 4 ⊢ (𝐴 = ∅ → (𝐴 ∈ On ↔ ∅ ∈ On))
14	12, 13	mpbiri 258	. . 3 ⊢ (𝐴 = ∅ → 𝐴 ∈ On)
15		onsuc 7790	. . . . 5 ⊢ (𝑥 ∈ On → suc 𝑥 ∈ On)
16		eleq1 2817	. . . . 5 ⊢ (𝐴 = suc 𝑥 → (𝐴 ∈ On ↔ suc 𝑥 ∈ On))
17	15, 16	syl5ibrcom 247	. . . 4 ⊢ (𝑥 ∈ On → (𝐴 = suc 𝑥 → 𝐴 ∈ On))
18	17	rexlimiv 3128	. . 3 ⊢ (∃𝑥 ∈ On 𝐴 = suc 𝑥 → 𝐴 ∈ On)
19		limelon 6400	. . 3 ⊢ ((𝐴 ∈ V ∧ Lim 𝐴) → 𝐴 ∈ On)
20	14, 18, 19	3jaoi 1430	. 2 ⊢ ((𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥 ∨ (𝐴 ∈ V ∧ Lim 𝐴)) → 𝐴 ∈ On)
21	11, 20	impbii 209	1 ⊢ (𝐴 ∈ On ↔ (𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥 ∨ (𝐴 ∈ V ∧ Lim 𝐴)))

Syntax hints:   ↔ wb 206   ∧ wa 395   ∨ w3o 1085   = wceq 1540   ∈ wcel 2109  ∃wrex 3054  Vcvv 3450  ∅c0 4299  Ord word 6334  Oncon0 6335  Lim wlim 6336  suc csuc 6337

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-11 2158  ax-ext 2702  ax-sep 5254  ax-nul 5264  ax-pr 5390  ax-un 7714

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-ne 2927  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-pss 3937  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-br 5111  df-opab 5173  df-tr 5218  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5594  df-we 5596  df-ord 6338  df-on 6339  df-lim 6340  df-suc 6341

This theorem is referenced by:  oawordeulem  8521  r1pwss  9744  r1val1  9746  pwcfsdom  10543  winalim2  10656  rankcf  10737  dfrdg4  35946  naddwordnexlem4  43397