Theorem ordsson 4309

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 4210	. . 3 |- ( ( Ord A /\ x e. A ) -> x e. On )
2	1	ex 113	. 2 |- ( Ord A -> ( x e. A -> x e. On ) )
3	2	ssrdv 3031	1 |- ( Ord A -> A C_ On )

Syntax hints:   -> wi 4   e. wcel 1438   C_ wss 2999   Ord word 4189   Oncon0 4190

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070

This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-in 3005  df-ss 3012  df-uni 3654  df-tr 3937  df-iord 4193  df-on 4195

This theorem is referenced by:  onss  4310  orduni  4312  iordsmo  6062  tfrlemi14d  6098  tfr1onlemssrecs  6104  tfri1dALT  6116  tfrcllemssrecs  6117  ordiso2  6728