Theorem elon2 4423

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 4422	. . 3 |- ( A e. On -> Ord A )
2		elex 2783	. . 3 |- ( A e. On -> A e. _V )
3	1, 2	jca 306	. 2 |- ( A e. On -> ( Ord A /\ A e. _V ) )
4		elong 4420	. . 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 2176   _Vcvv 2772   Ord word 4409   Oncon0 4410

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-rex 2490  df-v 2774  df-in 3172  df-ss 3179  df-uni 3851  df-tr 4143  df-iord 4413  df-on 4415

This theorem is referenced by:  tfrexlem  6420  pw1on  7338