Theorem ordsson 4558

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 4448	. . 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 3207	1 |- ( Ord A -> A C_ On )

Syntax hints:   -> wi 4   e. wcel 2178   C_ wss 3174   Ord word 4427   Oncon0 4428

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ral 2491  df-rex 2492  df-v 2778  df-in 3180  df-ss 3187  df-uni 3865  df-tr 4159  df-iord 4431  df-on 4433

This theorem is referenced by:  onss  4559  orduni  4561  iordsmo  6406  tfrlemi14d  6442  tfr1onlemssrecs  6448  tfri1dALT  6460  tfrcllemssrecs  6461  ordiso2  7163