Theorem ordeleqon 2980

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 2979	. . . 4 |- -. On e. V
2		elisset 1808	. . . 4 |- (On e. A -> On e. V)
3	1, 2	mto 106	. . 3 |- -. On e. A
4		ordon 2977	. . . . . 6 |- Ord On
5		ordtri3or 2969	. . . . . 6 |- ((Ord A /\ Ord On) -> (A e. On \/ A = On \/ On e. A))
6	4, 5	mpan2 694	. . . . 5 |- (Ord A -> (A e. On \/ A = On \/ On e. A))
7		df-3or 774	. . . . 5 |- ((A e. On \/ A = On \/ On e. A) <-> ((A e. On \/ A = On) \/ On e. A))
8	6, 7	sylib 198	. . . 4 |- (Ord A -> ((A e. On \/ A = On) \/ On e. A))
9	8	ord 232	. . 3 |- (Ord A -> (-. (A e. On \/ A = On) -> On e. A))
10	3, 9	mt3i 113	. 2 |- (Ord A -> (A e. On \/ A = On))
11		eloni 2948	. . 3 |- (A e. On -> Ord A)
12		ordeq 2945	. . . 4 |- (A = On -> (Ord A <-> Ord On))
13	4, 12	mpbiri 194	. . 3 |- (A = On -> Ord A)
14	11, 13	jaoi 341	. 2 |- ((A e. On \/ A = On) -> Ord A)
15	10, 14	impbi 157	1 |- (Ord A <-> (A e. On \/ A = On))

Syntax hints:   <-> wb 146   \/ wo 222   \/ w3o 772   = wceq 953   e. wcel 955  Vcvv 1802  Ord word 2937  Oncon0 2938

This theorem is referenced by:  ordsson 2981  ordunisuc 3079  orduninsuc 3104  limomss 3127  omon 3133  limom 3136  tfrlem13 3908  unialeph 4867

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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