Theorem elon2 4467

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 4466	. . 3 |- ( A e. On -> Ord A )
2		elex 2811	. . 3 |- ( A e. On -> A e. _V )
3	1, 2	jca 306	. 2 |- ( A e. On -> ( Ord A /\ A e. _V ) )
4		elong 4464	. . 3 |- ( A e. _V -> ( A e. On <-> Ord A ) )
5	4	biimparc 299	. 2 |- ( ( Ord A /\ A e. _V ) -> A e. On )
6	3, 5	impbii 126	1 |- ( A e. On <-> ( Ord A /\ A e. _V ) )

Syntax hints:   /\ wa 104   <-> wb 105   e. wcel 2200   _Vcvv 2799   Ord word 4453   Oncon0 4454

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2801  df-in 3203  df-ss 3210  df-uni 3889  df-tr 4183  df-iord 4457  df-on 4459

This theorem is referenced by:  tfrexlem  6480  pw1on  7411