Theorem elon 4896

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 4895	. 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 1819   _Vcvv 3109   Ord word 4886   Oncon0 4887

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435

This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ral 2812  df-rex 2813  df-v 3111  df-in 3478  df-ss 3485  df-uni 4252  df-tr 4551  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891

This theorem is referenced by:  tron  4910  0elon  4940  smogt  7056  rdglim2  7116  omeulem1  7249  isfinite2  7796  r0weon  8407  cflim3  8659  inar1  9170  ellimits  29765  dford3lem2  31173