Theorem elon2 4479

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 4478	. . 3 |- ( A e. On -> Ord A )
2		elex 2815	. . 3 |- ( A e. On -> A e. _V )
3	1, 2	jca 306	. 2 |- ( A e. On -> ( Ord A /\ A e. _V ) )
4		elong 4476	. . 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 2202   _Vcvv 2803   Ord word 4465   Oncon0 4466

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rex 2517  df-v 2805  df-in 3207  df-ss 3214  df-uni 3899  df-tr 4193  df-iord 4469  df-on 4471

This theorem is referenced by:  tfrexlem  6543  pw1on  7504