Theorem limelon 4377

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 4373	. . 3 |- ( Lim A -> Ord A )
2		elong 4351	. . 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 2136   Ord word 4340   Oncon0 4341   Lim wlim 4342

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 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-v 2728  df-in 3122  df-ss 3129  df-uni 3790  df-tr 4081  df-iord 4344  df-on 4346  df-ilim 4347

This theorem is referenced by: (None)