Theorem ordsson 4590

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 4480	. . 3 |- ( ( Ord A /\ x e. A ) -> x e. On )
2	1	ex 115	. 2 |- ( Ord A -> ( x e. A -> x e. On ) )
3	2	ssrdv 3233	1 |- ( Ord A -> A C_ On )

Syntax hints:   -> wi 4   e. wcel 2202   C_ wss 3200   Ord word 4459   Oncon0 4460

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-v 2804  df-in 3206  df-ss 3213  df-uni 3894  df-tr 4188  df-iord 4463  df-on 4465

This theorem is referenced by:  onss  4591  orduni  4593  iordsmo  6462  tfrlemi14d  6498  tfr1onlemssrecs  6504  tfri1dALT  6516  tfrcllemssrecs  6517  ordiso2  7233