Theorem elong 4421

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 4420	. 2 |- ( x = A -> ( Ord x <-> Ord A ) )
2		df-on 4416	. 2 |- On = { x | Ord x }
3	1, 2	elab2g 2920	1 |- ( A e. V -> ( A e. On <-> Ord A ) )

Syntax hints:   -> wi 4   <-> wb 105   e. wcel 2176   Ord word 4410   Oncon0 4411

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-rex 2490  df-v 2774  df-in 3172  df-ss 3179  df-uni 3851  df-tr 4144  df-iord 4414  df-on 4416

This theorem is referenced by:  elon  4422  eloni  4423  elon2  4424  ordelon  4431  onin  4434  limelon  4447  ssonuni  4537  onsuc  4550  onsucb  4552  onintonm  4566  onprc  4601  omelon2  4657  bj-nnelon  15932