Theorem limelon 4464

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 4460	. . 3 |- ( Lim A -> Ord A )
2		elong 4438	. . 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 2178   Ord word 4427   Oncon0 4428   Lim wlim 4429

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ral 2491  df-rex 2492  df-v 2778  df-in 3180  df-ss 3187  df-uni 3865  df-tr 4159  df-iord 4431  df-on 4433  df-ilim 4434

This theorem is referenced by: (None)