Theorem limelon 4496

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 4492	. . 3 |- ( Lim A -> Ord A )
2		elong 4470	. . 3 |- ( A e. B -> ( A e. On <-> Ord A ) )
3	1, 2	imbitrrid 156	. 2 |- ( A e. B -> ( Lim A -> A e. On ) )
4	3	imp 124	1 |- ( ( A e. B /\ Lim A ) -> A e. On )

Syntax hints:   -> wi 4   /\ wa 104   e. wcel 2202   Ord word 4459   Oncon0 4460   Lim wlim 4461

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-v 2804  df-in 3206  df-ss 3213  df-uni 3894  df-tr 4188  df-iord 4463  df-on 4465  df-ilim 4466

This theorem is referenced by: (None)