Theorem nlimon 3122

Description: Two ways to express 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:   x,y

Proof of Theorem nlimon

Step	Hyp	Ref	Expression
1		eloni 2958	. . 3 |- (x e. On -> Ord x)
2		dflim3 3118	. . . . 5 |- (Lim x <-> (Ord x /\ -. (x = (/) \/ E.y e. On x = suc y)))
3	2	baib 685	. . . 4 |- (Ord x -> (Lim x <-> -. (x = (/) \/ E.y e. On x = suc y)))
4	3	con2bid 526	. . 3 |- (Ord x -> ((x = (/) \/ E.y e. On x = suc y) <-> -. Lim x))
5	1, 4	syl 10	. 2 |- (x e. On -> ((x = (/) \/ E.y e. On x = suc y) <-> -. Lim x))
6	5	rabbii 1805	1 |- {x e. On | (x = (/) \/ E.y e. On x = suc y)} = {x e. On | -. Lim x}

Syntax hints:  -. wn 2   <-> wb 146   \/ wo 222   = wceq 956   e. wcel 958  E.wrex 1646  {crab 1648  (/)c0 2280  Ord word 2947  Oncon0 2948  Lim wlim 2949  suc csuc 2950

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2703  ax-nul 2710  ax-pow 2742  ax-pr 2779  ax-un 2866

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 776  df-3an 777  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-ral 1649  df-rex 1650  df-rab 1652  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-if 2362  df-pw 2402  df-sn 2412  df-pr 2413  df-tp 2415  df-op 2416  df-uni 2504  df-br 2620  df-opab 2667  df-tr 2681  df-eprel 2832  df-po 2840  df-so 2850  df-fr 2917  df-we 2934  df-ord 2951  df-on 2952  df-lim 2953  df-suc 2954

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