Theorem elon 5431

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 5430	. 2 |- ( A e. _V -> ( A e. On <-> Ord A ) )
3	1, 2	ax-mp 5	1 |- ( A e. On <-> Ord A )

Syntax hints:   <-> wb 188   e. wcel 1886   _Vcvv 3044   Ord word 5421   Oncon0 5422

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1668  ax-4 1681  ax-5 1757  ax-6 1804  ax-7 1850  ax-10 1914  ax-11 1919  ax-12 1932  ax-13 2090  ax-ext 2430

This theorem depends on definitions:  df-bi 189  df-an 373  df-tru 1446  df-ex 1663  df-nf 1667  df-sb 1797  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2580  df-ral 2741  df-rex 2742  df-v 3046  df-in 3410  df-ss 3417  df-uni 4198  df-tr 4497  df-po 4754  df-so 4755  df-fr 4792  df-we 4794  df-ord 5425  df-on 5426

This theorem is referenced by:  tron  5445  0elon  5475  smogt  7083  dfrecs3  7088  rdglim2  7147  omeulem1  7280  isfinite2  7826  r0weon  8440  cflim3  8689  inar1  9197  ellimits  30670  dford3lem2  35876