Theorem ordsson 2981

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 2977	. . . 4 |- Ord On
2		ordelssne 2964	. . . 4 |- ((Ord A /\ Ord On) -> (A e. On <-> (A (_ On /\ A =/= On)))
3	1, 2	mpan2 694	. . 3 |- (Ord A -> (A e. On <-> (A (_ On /\ A =/= On)))
4		pm3.26 319	. . 3 |- ((A (_ On /\ A =/= On) -> A (_ On)
5	3, 4	syl6bi 214	. 2 |- (Ord A -> (A e. On -> A (_ On))
6		ordeleqon 2980	. . . . 5 |- (Ord A <-> (A e. On \/ A = On))
7	6	biimp 151	. . . 4 |- (Ord A -> (A e. On \/ A = On))
8	7	ord 232	. . 3 |- (Ord A -> (-. A e. On -> A = On))
9		eqimss 2099	. . 3 |- (A = On -> A (_ On)
10	8, 9	syl6 22	. 2 |- (Ord A -> (-. A e. On -> A (_ On))
11	5, 10	pm2.61d 127	1 |- (Ord A -> A (_ On)

Syntax hints:  -. wn 2   -> wi 3   <-> wb 146   \/ wo 222   /\ wa 223   = wceq 953   e. wcel 955   =/= wne 1577   (_ wss 2037  Ord word 2937  Oncon0 2938

This theorem is referenced by:  onsst 2982  orduni 2987  ordsucuni 3076

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-sep 2693  ax-pow 2732  ax-pr 2769  ax-un 2857

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 774  df-3an 775  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-ral 1641  df-rex 1642  df-v 1803  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-nul 2271  df-pw 2392  df-sn 2402  df-pr 2403  df-tp 2405  df-op 2406  df-uni 2494  df-br 2610  df-opab 2657  df-tr 2671  df-eprel 2821  df-po 2831  df-so 2841  df-fr 2907  df-we 2924  df-ord 2941  df-on 2942

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