Theorem limelon 4384

Description: A limit ordinal class that is also a set is an ordinal number. (Contributed by NM, 26-Apr-2004.)

Assertion

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

Proof of Theorem limelon

Step	Hyp	Ref	Expression
1		limord 4380	. . 3 |- ( Lim A -> Ord A )
2		elong 4358	. . 3 |- ( A e. B -> ( A e. On <-> Ord A ) )
3	1, 2	syl5ibr 155	. 2 |- ( A e. B -> ( Lim A -> A e. On ) )
4	3	imp 123	1 |- ( ( A e. B /\ Lim A ) -> A e. On )

Syntax hints:   -> wi 4   /\ wa 103   e. wcel 2141   Ord word 4347   Oncon0 4348   Lim wlim 4349

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-in 3127  df-ss 3134  df-uni 3797  df-tr 4088  df-iord 4351  df-on 4353  df-ilim 4354

This theorem is referenced by: (None)