Theorem ordeleqon 2990

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 2989	. . . 4 |- -. On e. V
2		elisset 1817	. . . 4 |- (On e. A -> On e. V)
3	1, 2	mto 106	. . 3 |- -. On e. A
4		ordon 2987	. . . . . 6 |- Ord On
5		ordtri3or 2979	. . . . . 6 |- ((Ord A /\ Ord On) -> (A e. On \/ A = On \/ On e. A))
6	4, 5	mpan2 696	. . . . 5 |- (Ord A -> (A e. On \/ A = On \/ On e. A))
7		df-3or 776	. . . . 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 2958	. . 3 |- (A e. On -> Ord A)
12		ordeq 2955	. . . 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 774   = wceq 956   e. wcel 958  Vcvv 1811  Ord word 2947  Oncon0 2948

This theorem is referenced by:  ordsson 2991  ordunisuc 3089  orduninsuc 3114  limomss 3137  omon 3143  limom 3146  tfrlem13 3923  unialeph 4895

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2703  ax-pow 2742  ax-pr 2779  ax-un 2866

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 776  df-3an 777  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-ral 1649  df-rex 1650  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-tp 2415  df-op 2416  df-uni 2504  df-br 2620  df-opab 2667  df-tr 2681  df-eprel 2832  df-po 2840  df-so 2850  df-fr 2917  df-we 2934  df-ord 2951  df-on 2952

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