Theorem limelon 3038

Description: A limit ordinal class that is also a set is an ordinal number.

Assertion

Ref	Expression
limelon	|- ((A e. B /\ Lim A) -> A e. On)

Proof of Theorem limelon

Step	Hyp	Ref	Expression
1		elong 2962	. . 3 |- (A e. B -> (A e. On <-> Ord A))
2		limord 3034	. . 3 |- (Lim A -> Ord A)
3	1, 2	syl5bir 210	. 2 |- (A e. B -> (Lim A -> A e. On))
4	3	imp 350	1 |- ((A e. B /\ Lim A) -> A e. On)

Syntax hints:   -> wi 3   /\ wa 223   e. wcel 960  Ord word 2953  Oncon0 2954  Lim wlim 2955

This theorem is referenced by:  limuni3 3129  dfom2 3139  tfindsg2 3169  rdglimt 3954  oalim 4173  omlim 4174  oelim 4175  oalimcl 4200  oaass 4201  omlimcl 4215  odi 4216  omass 4217  oen0 4219  oewordri 4225  oelim2 4228  r1pwcl 4697  alephordi 4885  cflim 4921

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-12 970  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 779  df-ex 983  df-sb 1174  df-clab 1467  df-cleq 1472  df-clel 1475  df-ral 1652  df-rex 1653  df-v 1815  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-sn 2416  df-pr 2417  df-op 2420  df-uni 2508  df-br 2625  df-tr 2686  df-po 2846  df-so 2856  df-fr 2923  df-we 2940  df-ord 2957  df-on 2958  df-lim 2959

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