Theorem dflim3 2368

Description: An alternate definition of a limit ordinal, which is any ordinal that is neither zero nor a successor.

Assertion

Ref	Expression
dflim3	⊢ (Lim A ↔ (Ord A ∧ ¬ (A = ∅ ∨ ∃x ∈ On A = suc x)))

Distinct variable group(s):   x,A

Proof of Theorem dflim3

Step	Hyp	Ref	Expression
1		limord 2283	. . 3 ⊢ (Lim A → Ord A)
2		nlim0 2282	. . . . . . 7 ⊢ ¬ Lim ∅
3		limeq 2211	. . . . . . 7 ⊢ (A = ∅ → (Lim A ↔ Lim ∅))
4	2, 3	mtbiri 539	. . . . . 6 ⊢ (A = ∅ → ¬ Lim A)
5	4	con2i 89	. . . . 5 ⊢ (Lim A → ¬ A = ∅)
6		limuni 2284	. . . . . 6 ⊢ (Lim A → A = ∪A)
7		orduninsuc 2365	. . . . . . 7 ⊢ (Ord A → (A = ∪A ↔ ¬ ∃x ∈ On A = suc x))
8	1, 7	syl 12	. . . . . 6 ⊢ (Lim A → (A = ∪A ↔ ¬ ∃x ∈ On A = suc x))
9	6, 8	mpbid 170	. . . . 5 ⊢ (Lim A → ¬ ∃x ∈ On A = suc x)
10	5, 9	jca 236	. . . 4 ⊢ (Lim A → (¬ A = ∅ ∧ ¬ ∃x ∈ On A = suc x))
11		ioran 254	. . . 4 ⊢ (¬ (A = ∅ ∨ ∃x ∈ On A = suc x) ↔ (¬ A = ∅ ∧ ¬ ∃x ∈ On A = suc x))
12	10, 11	sylibr 175	. . 3 ⊢ (Lim A → ¬ (A = ∅ ∨ ∃x ∈ On A = suc x))
13	1, 12	jca 236	. 2 ⊢ (Lim A → (Ord A ∧ ¬ (A = ∅ ∨ ∃x ∈ On A = suc x)))
14		ordzsl 2366	. . . . 5 ⊢ (Ord A ↔ (A = ∅ ∨ ∃x ∈ On A = suc x ∨ Lim A))
15	14	biimp 133	. . . 4 ⊢ (Ord A → (A = ∅ ∨ ∃x ∈ On A = suc x ∨ Lim A))
16		df-3or 582	. . . 4 ⊢ ((A = ∅ ∨ ∃x ∈ On A = suc x ∨ Lim A) ↔ ((A = ∅ ∨ ∃x ∈ On A = suc x) ∨ Lim A))
17	15, 16	sylib 173	. . 3 ⊢ (Ord A → ((A = ∅ ∨ ∃x ∈ On A = suc x) ∨ Lim A))
18	17	orcanai 515	. 2 ⊢ ((Ord A ∧ ¬ (A = ∅ ∨ ∃x ∈ On A = suc x)) → Lim A)
19	13, 18	impbi 139	1 ⊢ (Lim A ↔ (Ord A ∧ ¬ (A = ∅ ∨ ∃x ∈ On A = suc x)))

Syntax hints:  ¬ wn 1   ↔ wb 127   ∨ wo 195   ∧ wa 196   ∨ w3o 580   = wceq 1091  ∃wrex 1202  ∅c0 1707  ∪cuni 1919  Ord word 2198  Oncon0 2199  Lim wlim 2200  suc csuc 2201

This theorem is referenced by:  nlimon 2369  oalimcl 3162

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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