Theorem elon2 4348

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 4347	. . 3 |- ( A e. On -> Ord A )
2		elex 2732	. . 3 |- ( A e. On -> A e. _V )
3	1, 2	jca 304	. 2 |- ( A e. On -> ( Ord A /\ A e. _V ) )
4		elong 4345	. . 3 |- ( A e. _V -> ( A e. On <-> Ord A ) )
5	4	biimparc 297	. 2 |- ( ( Ord A /\ A e. _V ) -> A e. On )
6	3, 5	impbii 125	1 |- ( A e. On <-> ( Ord A /\ A e. _V ) )

Syntax hints:   /\ wa 103   <-> wb 104   e. wcel 2135   _Vcvv 2721   Ord word 4334   Oncon0 4335

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-ext 2146

This theorem depends on definitions:  df-bi 116  df-tru 1345  df-nf 1448  df-sb 1750  df-clab 2151  df-cleq 2157  df-clel 2160  df-nfc 2295  df-ral 2447  df-rex 2448  df-v 2723  df-in 3117  df-ss 3124  df-uni 3784  df-tr 4075  df-iord 4338  df-on 4340

This theorem is referenced by:  tfrexlem  6293  pw1on  7173