Theorem ordsson 2242

Description: Any ordinal class is a subclass of the class of ordinal numbers. Corollary 7.15 of [TakeutiZaring] p. 38.

Assertion

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

Proof of Theorem ordsson

Step	Hyp	Ref	Expression
1		ordon 2238	. . . 4 |- Ord On
2		ordelssne 2225	. . . 4 |- ((Ord A /\ Ord On) -> (A e. On <-> (A (_ On /\ -. A = On)))
3	1, 2	mpan2 519	. . 3 |- (Ord A -> (A e. On <-> (A (_ On /\ -. A = On)))
4		pm3.26 256	. . 3 |- ((A (_ On /\ -. A = On) -> A (_ On)
5	3, 4	syl6bi 187	. 2 |- (Ord A -> (A e. On -> A (_ On))
6		ordeleqon 2241	. . . . 5 |- (Ord A <-> (A e. On \/ A = On))
7	6	biimp 133	. . . 4 |- (Ord A -> (A e. On \/ A = On))
8	7	ord 202	. . 3 |- (Ord A -> (-. A e. On -> A = On))
9		eqimss 1548	. . 3 |- (A = On -> A (_ On)
10	8, 9	syl6 23	. 2 |- (Ord A -> (-. A e. On -> A (_ On))
11	5, 10	pm2.61d 112	1 |- (Ord A -> A (_ On)

Syntax hints:  -. wn 1   -> wi 2   <-> wb 127   \/ wo 195   /\ wa 196   = wceq 1091   e. wcel 1092   (_ wss 1487  Ord word 2198  Oncon0 2199

This theorem is referenced by:  onsst 2243  uniord 2252  ordsucuni 2336

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3or 582  df-3an 583  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-rex 1206  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-tp 1814  df-op 1815  df-uni 1920  df-tr 2042  df-br 2063  df-opab 2098  df-eprel 2122  df-po 2128  df-so 2138  df-fr 2169  df-we 2186  df-ord 2202  df-on 2203

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