Theorem elon2 4407

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 4406	. . 3 |- ( A e. On -> Ord A )
2		elex 2771	. . 3 |- ( A e. On -> A e. _V )
3	1, 2	jca 306	. 2 |- ( A e. On -> ( Ord A /\ A e. _V ) )
4		elong 4404	. . 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 2164   _Vcvv 2760   Ord word 4393   Oncon0 4394

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-in 3159  df-ss 3166  df-uni 3836  df-tr 4128  df-iord 4397  df-on 4399

This theorem is referenced by:  tfrexlem  6387  pw1on  7286