Theorem ordeleqon 2241

Description: A way to express the ordinal property of a class in terms of the class of ordinal numbers. Corollary 7.14 of [TakeutiZaring] p. 38 and its converse.

Assertion

Ref	Expression
ordeleqon	|- (Ord A <-> (A e. On \/ A = On))

Proof of Theorem ordeleqon

Step	Hyp	Ref	Expression
1		onprc 2240	. . . 4 |- -. On e. V
2		elisset 1354	. . . 4 |- (On e. A -> On e. V)
3	1, 2	mto 93	. . 3 |- -. On e. A
4		ordon 2238	. . . . . 6 |- Ord On
5		ordtri3or 2230	. . . . . 6 |- ((Ord A /\ Ord On) -> (A e. On \/ A = On \/ On e. A))
6	4, 5	mpan2 519	. . . . 5 |- (Ord A -> (A e. On \/ A = On \/ On e. A))
7		df-3or 582	. . . . 5 |- ((A e. On \/ A = On \/ On e. A) <-> ((A e. On \/ A = On) \/ On e. A))
8	6, 7	sylib 173	. . . 4 |- (Ord A -> ((A e. On \/ A = On) \/ On e. A))
9	8	ord 202	. . 3 |- (Ord A -> (-. (A e. On \/ A = On) -> On e. A))
10	3, 9	mt3i 100	. 2 |- (Ord A -> (A e. On \/ A = On))
11		eloni 2209	. . 3 |- (A e. On -> Ord A)
12		ordeq 2206	. . . 4 |- (A = On -> (Ord A <-> Ord On))
13	4, 12	mpbiri 169	. . 3 |- (A = On -> Ord A)
14	11, 13	jaoi 275	. 2 |- ((A e. On \/ A = On) -> Ord A)
15	10, 14	impbi 139	1 |- (Ord A <-> (A e. On \/ A = On))

Syntax hints:   <-> wb 127   \/ wo 195   \/ w3o 580   = wceq 1091   e. wcel 1092  Vcvv 1348  Ord word 2198  Oncon0 2199

This theorem is referenced by:  ordsson 2242  ordunisuc 2339  limsuc 2361  orduninsuc 2365  limomss 2378  omon 2384  limom 2387  tfrlem13 2961

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

metamath.org