Theorem nlimon 2369

Description: Two ways of expressing the class of non-limit ordinal numbers. Part of Definition 7.27 of [TakeutiZaring] p. 42, who use the symbol K_I for this class.

Assertion

Ref	Expression
nlimon	|- {x e. On | (x = (/) \/ E.y e. On x = suc y)} = {x e. On | -. Lim x}

Distinct variable group(s):   x,y

Proof of Theorem nlimon

Step	Hyp	Ref	Expression
1		eloni 2209	. . 3 |- (x e. On -> Ord x)
2		dflim3 2368	. . . . 5 |- (Lim x <-> (Ord x /\ -. (x = (/) \/ E.y e. On x = suc y)))
3	2	baib 506	. . . 4 |- (Ord x -> (Lim x <-> -. (x = (/) \/ E.y e. On x = suc y)))
4	3	bicon2d 404	. . 3 |- (Ord x -> ((x = (/) \/ E.y e. On x = suc y) <-> -. Lim x))
5	1, 4	syl 12	. 2 |- (x e. On -> ((x = (/) \/ E.y e. On x = suc y) <-> -. Lim x))
6	5	birabi 1342	1 |- {x e. On | (x = (/) \/ E.y e. On x = suc y)} = {x e. On | -. Lim x}

Syntax hints:  -. wn 1   <-> wb 127   \/ wo 195   = wceq 1091   e. wcel 1092  E.wrex 1202  {crab 1204  (/)c0 1707  Ord word 2198  Oncon0 2199  Lim wlim 2200  suc csuc 2201

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-rab 1208  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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