Theorem onzsl 2367

Description: An ordinal number is zero, a successor ordinal, or a limit ordinal number.

Assertion

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

Distinct variable group(s):   x,A

Proof of Theorem onzsl

Step	Hyp	Ref	Expression
1		elisset 1354	. . 3 ⊢ (A ∈ On → A ∈ V)
2	1	pm4.71ri 484	. 2 ⊢ (A ∈ On ↔ (A ∈ V ∧ A ∈ On))
3		elong 2207	. . 3 ⊢ (A ∈ V → (A ∈ On ↔ Ord A))
4	3	pm5.32i 489	. 2 ⊢ ((A ∈ V ∧ A ∈ On) ↔ (A ∈ V ∧ Ord A))
5		andi 456	. . . 4 ⊢ ((A ∈ V ∧ ((A = ∅ ∨ ∃x ∈ On A = suc x) ∨ Lim A)) ↔ ((A ∈ V ∧ (A = ∅ ∨ ∃x ∈ On A = suc x)) ∨ (A ∈ V ∧ Lim A)))
6		0ex 1745	. . . . . . . 8 ⊢ ∅ ∈ V
7		eleq1 1149	. . . . . . . 8 ⊢ (A = ∅ → (A ∈ V ↔ ∅ ∈ V))
8	6, 7	mpbiri 169	. . . . . . 7 ⊢ (A = ∅ → A ∈ V)
9		visset 1350	. . . . . . . . . . 11 ⊢ x ∈ V
10	9	sucex 2303	. . . . . . . . . 10 ⊢ suc x ∈ V
11		eleq1 1149	. . . . . . . . . 10 ⊢ (A = suc x → (A ∈ V ↔ suc x ∈ V))
12	10, 11	mpbiri 169	. . . . . . . . 9 ⊢ (A = suc x → A ∈ V)
13	12	a1i 7	. . . . . . . 8 ⊢ (x ∈ On → (A = suc x → A ∈ V))
14	13	r19.23aiv 1284	. . . . . . 7 ⊢ (∃x ∈ On A = suc x → A ∈ V)
15	8, 14	jaoi 275	. . . . . 6 ⊢ ((A = ∅ ∨ ∃x ∈ On A = suc x) → A ∈ V)
16	15	pm4.71ri 484	. . . . 5 ⊢ ((A = ∅ ∨ ∃x ∈ On A = suc x) ↔ (A ∈ V ∧ (A = ∅ ∨ ∃x ∈ On A = suc x)))
17	16	orbi1i 215	. . . 4 ⊢ (((A = ∅ ∨ ∃x ∈ On A = suc x) ∨ (A ∈ V ∧ Lim A)) ↔ ((A ∈ V ∧ (A = ∅ ∨ ∃x ∈ On A = suc x)) ∨ (A ∈ V ∧ Lim A)))
18	5, 17	bitr4 154	. . 3 ⊢ ((A ∈ V ∧ ((A = ∅ ∨ ∃x ∈ On A = suc x) ∨ Lim A)) ↔ ((A = ∅ ∨ ∃x ∈ On A = suc x) ∨ (A ∈ V ∧ Lim A)))
19		ordzsl 2366	. . . . 5 ⊢ (Ord A ↔ (A = ∅ ∨ ∃x ∈ On A = suc x ∨ Lim A))
20		df-3or 582	. . . . 5 ⊢ ((A = ∅ ∨ ∃x ∈ On A = suc x ∨ Lim A) ↔ ((A = ∅ ∨ ∃x ∈ On A = suc x) ∨ Lim A))
21	19, 20	bitr 151	. . . 4 ⊢ (Ord A ↔ ((A = ∅ ∨ ∃x ∈ On A = suc x) ∨ Lim A))
22	21	anbi2i 367	. . 3 ⊢ ((A ∈ V ∧ Ord A) ↔ (A ∈ V ∧ ((A = ∅ ∨ ∃x ∈ On A = suc x) ∨ Lim A)))
23		df-3or 582	. . 3 ⊢ ((A = ∅ ∨ ∃x ∈ On A = suc x ∨ (A ∈ V ∧ Lim A)) ↔ ((A = ∅ ∨ ∃x ∈ On A = suc x) ∨ (A ∈ V ∧ Lim A)))
24	18, 22, 23	3bitr4 158	. 2 ⊢ ((A ∈ V ∧ Ord A) ↔ (A = ∅ ∨ ∃x ∈ On A = suc x ∨ (A ∈ V ∧ Lim A)))
25	2, 4, 24	3bitr 155	1 ⊢ (A ∈ On ↔ (A = ∅ ∨ ∃x ∈ On A = suc x ∨ (A ∈ V ∧ Lim A)))

Syntax hints:   → wi 2   ↔ wb 127   ∨ wo 195   ∧ wa 196   ∨ w3o 580   = wceq 1091   ∈ wcel 1092  ∃wrex 1202  Vcvv 1348  ∅c0 1707  Ord word 2198  Oncon0 2199  Lim wlim 2200  suc csuc 2201

This theorem is referenced by:  oawordeulem 3156  r1val1 3502

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3or 582  df-3an 583  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-rex 1206  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-if 1777  df-pw 1799  df-sn 1811  df-pr 1812  df-tp 1814  df-op 1815  df-uni 1920  df-tr 2042  df-br 2063  df-opab 2098  df-eprel 2122  df-po 2128  df-so 2138  df-fr 2169  df-we 2186  df-ord 2202  df-on 2203  df-lim 2204  df-suc 2205

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