Theorem elon 4837

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 4836	. 2 |- ( A e. _V -> ( A e. On <-> Ord A ) )
3	1, 2	ax-mp 5	1 |- ( A e. On <-> Ord A )

Syntax hints:   <-> wb 184   e. wcel 1758   _Vcvv 3078   Ord word 4827   Oncon0 4828

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ral 2804  df-rex 2805  df-v 3080  df-in 3444  df-ss 3451  df-uni 4201  df-tr 4495  df-po 4750  df-so 4751  df-fr 4788  df-we 4790  df-ord 4831  df-on 4832

This theorem is referenced by:  tron  4851  0elon  4881  smogt  6939  rdglim2  6999  omeulem1  7132  isfinite2  7682  r0weon  8291  cflim3  8543  inar1  9054  ellimits  28086  dford3lem2  29525