Theorem elong 4128

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

Assertion

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

Proof of Theorem elong

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ordeq 4127	. 2 |- ( x = A -> ( Ord x <-> Ord A ) )
2		df-on 4123	. 2 |- On = { x | Ord x }
3	1, 2	elab2g 2740	1 |- ( A e. V -> ( A e. On <-> Ord A ) )

Syntax hints:   -> wi 4   <-> wb 103   e. wcel 1433   Ord word 4117   Oncon0 4118

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-v 2603  df-in 2979  df-ss 2986  df-uni 3602  df-tr 3876  df-iord 4121  df-on 4123

This theorem is referenced by:  elon  4129  eloni  4130  elon2  4131  ordelon  4138  onin  4141  limelon  4154  ssonuni  4232  suceloni  4245  sucelon  4247  onintonm  4261  onprc  4295  omelon2  4348  bj-nnelon  10754