Theorem elon2 4354

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 4353	. . 3 |- ( A e. On -> Ord A )
2		elex 2737	. . 3 |- ( A e. On -> A e. _V )
3	1, 2	jca 304	. 2 |- ( A e. On -> ( Ord A /\ A e. _V ) )
4		elong 4351	. . 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 2136   _Vcvv 2726   Ord word 4340   Oncon0 4341

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-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

This theorem is referenced by:  tfrexlem  6302  pw1on  7182