Theorem elon2 5734

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		elex 3212	. . 3 |- ( A e. On -> A e. _V )
2		elong 5731	. . 3 |- ( A e. _V -> ( A e. On <-> Ord A ) )
3	1, 2	biadan2 674	. 2 |- ( A e. On <-> ( A e. _V /\ Ord A ) )
4		ancom 466	. 2 |- ( ( A e. _V /\ Ord A ) <-> ( Ord A /\ A e. _V ) )
5	3, 4	bitri 264	1 |- ( A e. On <-> ( Ord A /\ A e. _V ) )

Syntax hints:   <-> wb 196   /\ wa 384   e. wcel 1990   _Vcvv 3200   Ord word 5722   Oncon0 5723

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-v 3202  df-in 3581  df-ss 3588  df-uni 4437  df-tr 4753  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-ord 5726  df-on 5727

This theorem is referenced by:  sucelon  7017  tfrlem12  7485  tfrlem13  7486  gruina  9640  bdayimaon  31843