Theorem elon2 4293

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 4292	. . 3 |- ( A e. On -> Ord A )
2		elex 2692	. . 3 |- ( A e. On -> A e. _V )
3	1, 2	jca 304	. 2 |- ( A e. On -> ( Ord A /\ A e. _V ) )
4		elong 4290	. . 3 |- ( A e. _V -> ( A e. On <-> Ord A ) )
5	4	biimparc 297	. 2 |- ( ( Ord A /\ A e. _V ) -> A e. On )
6	3, 5	impbii 125	1 |- ( A e. On <-> ( Ord A /\ A e. _V ) )

Syntax hints:   /\ wa 103   <-> wb 104   e. wcel 1480   _Vcvv 2681   Ord word 4279   Oncon0 4280

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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-v 2683  df-in 3072  df-ss 3079  df-uni 3732  df-tr 4022  df-iord 4283  df-on 4285

This theorem is referenced by:  tfrexlem  6224