Theorem ordsson 4474

Description: Any ordinal class is a subclass of the class of ordinal numbers. Corollary 7.15 of [TakeutiZaring] p. 38. (Contributed by NM, 18-May-1994.)

Assertion

Ref	Expression
ordsson	|- ( Ord A -> A C_ On )

Proof of Theorem ordsson

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ordelon 4366	. . 3 |- ( ( Ord A /\ x e. A ) -> x e. On )
2	1	ex 114	. 2 |- ( Ord A -> ( x e. A -> x e. On ) )
3	2	ssrdv 3153	1 |- ( Ord A -> A C_ On )

Syntax hints:   -> wi 4   e. wcel 2141   C_ wss 3121   Ord word 4345   Oncon0 4346

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-3an 975  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 3795  df-tr 4086  df-iord 4349  df-on 4351

This theorem is referenced by:  onss  4475  orduni  4477  iordsmo  6274  tfrlemi14d  6310  tfr1onlemssrecs  6316  tfri1dALT  6328  tfrcllemssrecs  6329  ordiso2  7009