Theorem elong 4358

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 4357	. 2 |- ( x = A -> ( Ord x <-> Ord A ) )
2		df-on 4353	. 2 |- On = { x | Ord x }
3	1, 2	elab2g 2877	1 |- ( A e. V -> ( A e. On <-> Ord A ) )

Syntax hints:   -> wi 4   <-> wb 104   e. wcel 2141   Ord word 4347   Oncon0 4348

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-in 3127  df-ss 3134  df-uni 3797  df-tr 4088  df-iord 4351  df-on 4353

This theorem is referenced by:  elon  4359  eloni  4360  elon2  4361  ordelon  4368  onin  4371  limelon  4384  ssonuni  4472  suceloni  4485  sucelon  4487  onintonm  4501  onprc  4536  omelon2  4592  bj-nnelon  13994