Theorem elon2 4160

Description: An ordinal number is an ordinal set. (Contributed by NM, 8-Feb-2004.)

Assertion

Ref	Expression
elon2	|- ( A e. On <-> ( Ord A /\ A e. _V ) )

Proof of Theorem elon2

Step	Hyp	Ref	Expression
1		eloni 4159	. . 3 |- ( A e. On -> Ord A )
2		elex 2619	. . 3 |- ( A e. On -> A e. _V )
3	1, 2	jca 300	. 2 |- ( A e. On -> ( Ord A /\ A e. _V ) )
4		elong 4157	. . 3 |- ( A e. _V -> ( A e. On <-> Ord A ) )
5	4	biimparc 293	. 2 |- ( ( Ord A /\ A e. _V ) -> A e. On )
6	3, 5	impbii 124	1 |- ( A e. On <-> ( Ord A /\ A e. _V ) )

Syntax hints:   /\ wa 102   <-> wb 103   e. wcel 1434   _Vcvv 2610   Ord word 4146   Oncon0 4147

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065

This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rex 2359  df-v 2612  df-in 2989  df-ss 2996  df-uni 3623  df-tr 3897  df-iord 4150  df-on 4152

This theorem is referenced by:  tfrexlem  6004