Theorem elon2 4113

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 4112	. . 3 |- ( A e. On -> Ord A )
2		elex 2566	. . 3 |- ( A e. On -> A e. _V )
3	1, 2	jca 290	. 2 |- ( A e. On -> ( Ord A /\ A e. _V ) )
4		elong 4110	. . 3 |- ( A e. _V -> ( A e. On <-> Ord A ) )
5	4	biimparc 283	. 2 |- ( ( Ord A /\ A e. _V ) -> A e. On )
6	3, 5	impbii 117	1 |- ( A e. On <-> ( Ord A /\ A e. _V ) )

Syntax hints:   /\ wa 97   <-> wb 98   e. wcel 1393   _Vcvv 2557   Ord word 4099   Oncon0 4100

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-in 2924  df-ss 2931  df-uni 3581  df-tr 3855  df-iord 4103  df-on 4105

This theorem is referenced by:  tfrexlem  5948