Theorem elon 4368

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 4367	. 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 2146   _Vcvv 2735   Ord word 4356   Oncon0 4357

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 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-ext 2157

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1459  df-sb 1761  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ral 2458  df-rex 2459  df-v 2737  df-in 3133  df-ss 3140  df-uni 3806  df-tr 4097  df-iord 4360  df-on 4362

This theorem is referenced by:  tron  4376  0elon  4386  ordtriexmidlem  4512  ontr2exmid  4518  ordtri2or2exmidlem  4519  onsucelsucexmidlem  4522  exmidonfinlem  7182  bj-omelon  14253