Theorem ordeleqon 2996

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 2995	. . . 4 |- -. On e. V
2		elisset 1820	. . . 4 |- (On e. A -> On e. V)
3	1, 2	mto 106	. . 3 |- -. On e. A
4		ordon 2993	. . . . . 6 |- Ord On
5		ordtri3or 2985	. . . . . 6 |- ((Ord A /\ Ord On) -> (A e. On \/ A = On \/ On e. A))
6	4, 5	mpan2 698	. . . . 5 |- (Ord A -> (A e. On \/ A = On \/ On e. A))
7		df-3or 778	. . . . 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 2964	. . 3 |- (A e. On -> Ord A)
12		ordeq 2961	. . . 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 776   = wceq 958   e. wcel 960  Vcvv 1814  Ord word 2953  Oncon0 2954

This theorem is referenced by:  ordsson 2997  ordunisuc 3095  orduninsuc 3120  limomss 3143  omon 3149  limom 3152  tfrlem13 3929  unialeph 4906

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-sep 2708  ax-pow 2748  ax-pr 2785  ax-un 2872

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 778  df-3an 779  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-ral 1652  df-rex 1653  df-v 1815  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-pw 2406  df-sn 2416  df-pr 2417  df-tp 2419  df-op 2420  df-uni 2508  df-br 2625  df-opab 2672  df-tr 2686  df-eprel 2838  df-po 2846  df-so 2856  df-fr 2923  df-we 2940  df-ord 2957  df-on 2958

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