Theorem elong 4408

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 4407	. 2 |- ( x = A -> ( Ord x <-> Ord A ) )
2		df-on 4403	. 2 |- On = { x | Ord x }
3	1, 2	elab2g 2911	1 |- ( A e. V -> ( A e. On <-> Ord A ) )

Syntax hints:   -> wi 4   <-> wb 105   e. wcel 2167   Ord word 4397   Oncon0 4398

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-v 2765  df-in 3163  df-ss 3170  df-uni 3840  df-tr 4132  df-iord 4401  df-on 4403

This theorem is referenced by:  elon  4409  eloni  4410  elon2  4411  ordelon  4418  onin  4421  limelon  4434  ssonuni  4524  onsuc  4537  onsucb  4539  onintonm  4553  onprc  4588  omelon2  4644  bj-nnelon  15605