Theorem elon2 4361

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 4360	. . 3 |- ( A e. On -> Ord A )
2		elex 2741	. . 3 |- ( A e. On -> A e. _V )
3	1, 2	jca 304	. 2 |- ( A e. On -> ( Ord A /\ A e. _V ) )
4		elong 4358	. . 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 2141   _Vcvv 2730   Ord word 4347   Oncon0 4348

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-in 3127  df-ss 3134  df-uni 3797  df-tr 4088  df-iord 4351  df-on 4353

This theorem is referenced by:  tfrexlem  6313  pw1on  7203