Theorem ordeleqon 3866

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 3865	. . . 4 |- -. On e. _V
2		elisset 2299	. . . 4 |- (On e. A -> On e. _V)
3	1, 2	mto 121	. . 3 |- -. On e. A
4		ordon 3863	. . . . . 6 |- Ord On
5		ordtri3or 3691	. . . . . 6 |- ((Ord A /\ Ord On) -> (A e. On \/ A = On \/ On e. A))
6	4, 5	mpan2 760	. . . . 5 |- (Ord A -> (A e. On \/ A = On \/ On e. A))
7		df-3or 859	. . . . 5 |- ((A e. On \/ A = On \/ On e. A) <-> ((A e. On \/ A = On) \/ On e. A))
8	6, 7	sylib 215	. . . 4 |- (Ord A -> ((A e. On \/ A = On) \/ On e. A))
9	8	ord 249	. . 3 |- (Ord A -> (-. (A e. On \/ A = On) -> On e. A))
10	3, 9	mt3i 128	. 2 |- (Ord A -> (A e. On \/ A = On))
11		eloni 3667	. . 3 |- (A e. On -> Ord A)
12		ordeq 3664	. . . 4 |- (A = On -> (Ord A <-> Ord On))
13	4, 12	mpbiri 211	. . 3 |- (A = On -> Ord A)
14	11, 13	jaoi 368	. 2 |- ((A e. On \/ A = On) -> Ord A)
15	10, 14	impbii 174	1 |- (Ord A <-> (A e. On \/ A = On))

Syntax hints:   <-> wb 163   \/ wo 239   \/ w3o 857   = wceq 1298   e. wcel 1300  _Vcvv 2292  Ord word 3656  Oncon0 3657

This theorem is referenced by:  ordsson 3867  ordssonOLD 3868  ordunisuc 3911  ordunisucOLD 3912  orduninsuc 3925  limomss 3956  omon 3963  limom 3967  tfrlem13 5131  unialeph 6043

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3or 859  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-pss 2607  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-tp 3052  df-op 3053  df-uni 3178  df-br 3339  df-opab 3396  df-tr 3412  df-eprel 3583  df-po 3591  df-so 3604  df-fr 3625  df-we 3644  df-ord 3660  df-on 3661

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