Theorem limelon 3032

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 2956	. . 3 |- (A e. B -> (A e. On <-> Ord A))
2		limord 3028	. . 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 958  Ord word 2947  Oncon0 2948  Lim wlim 2949

This theorem is referenced by:  limuni3 3123  dfom2 3133  tfindsg2 3163  rdglimt 3948  oalim 4167  omlim 4168  oelim 4169  oalimcl 4194  oaass 4195  omlimcl 4209  odi 4210  omass 4211  oen0 4213  oewordri 4219  oelim2 4222  r1pwcl 4687  alephordi 4874  cflim 4909

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-12 968  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

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 777  df-ex 981  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-ral 1649  df-rex 1650  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-sn 2412  df-pr 2413  df-op 2416  df-uni 2504  df-br 2620  df-tr 2681  df-po 2840  df-so 2850  df-fr 2917  df-we 2934  df-ord 2951  df-on 2952  df-lim 2953

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