Theorem limelon 3027

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 2951	. . 3 |- (A e. B -> (A e. On <-> Ord A))
2		limord 3023	. . 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 956  Ord word 2942  Oncon0 2943  Lim wlim 2944

This theorem is referenced by:  limuni3 3118  dfom2 3128  tfindsg2 3158  rdglimt 3939  oalim 4157  omlim 4158  oelim 4159  oalimcl 4184  oaass 4185  omlimcl 4199  odi 4200  omass 4201  oen0 4203  oewordri 4209  oelim2 4212  r1pwcl 4667  alephordi 4854  cflim 4889

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-10 964  ax-12 966  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 776  df-ex 979  df-sb 1170  df-clab 1462  df-cleq 1467  df-clel 1470  df-ral 1646  df-rex 1647  df-v 1808  df-dif 2045  df-un 2046  df-in 2047  df-ss 2049  df-nul 2277  df-sn 2408  df-pr 2409  df-op 2412  df-uni 2499  df-br 2615  df-tr 2676  df-po 2835  df-so 2845  df-fr 2912  df-we 2929  df-ord 2946  df-on 2947  df-lim 2948

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