Theorem elon 4439

Description: An ordinal number is an ordinal set. (Contributed by NM, 5-Jun-1994.)

Hypothesis

Ref	Expression
elon.1	|- A e. _V

Assertion

Ref	Expression
elon	|- ( A e. On <-> Ord A )

Proof of Theorem elon

Step	Hyp	Ref	Expression
1		elon.1	. 2 |- A e. _V
2		elong 4438	. 2 |- ( A e. _V -> ( A e. On <-> Ord A ) )
3	1, 2	ax-mp 5	1 |- ( A e. On <-> Ord A )

Syntax hints:   <-> wb 105   e. wcel 2178   _Vcvv 2776   Ord word 4427   Oncon0 4428

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 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ral 2491  df-rex 2492  df-v 2778  df-in 3180  df-ss 3187  df-uni 3865  df-tr 4159  df-iord 4431  df-on 4433

This theorem is referenced by:  tron  4447  0elon  4457  ordtriexmidlem  4585  ontr2exmid  4591  ordtri2or2exmidlem  4592  onsucelsucexmidlem  4595  exmidonfinlem  7332  bj-omelon  16096