Theorem ordeleqon 6988

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. (Contributed by NM, 1-Jun-2003.)

Assertion

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

Proof of Theorem ordeleqon

Step	Hyp	Ref	Expression
1		onprc 6984	. . . 4 |- -. On e. _V
2		elex 3212	. . . 4 |- ( On e. A -> On e. _V )
3	1, 2	mto 188	. . 3 |- -. On e. A
4		ordon 6982	. . . . . 6 |- Ord On
5		ordtri3or 5755	. . . . . 6 |- ( ( Ord A /\ Ord On ) -> ( A e. On \/ A = On \/ On e. A ) )
6	4, 5	mpan2 707	. . . . 5 |- ( Ord A -> ( A e. On \/ A = On \/ On e. A ) )
7		df-3or 1038	. . . . 5 |- ( ( A e. On \/ A = On \/ On e. A ) <-> ( ( A e. On \/ A = On ) \/ On e. A ) )
8	6, 7	sylib 208	. . . 4 |- ( Ord A -> ( ( A e. On \/ A = On ) \/ On e. A ) )
9	8	ord 392	. . 3 |- ( Ord A -> ( -. ( A e. On \/ A = On ) -> On e. A ) )
10	3, 9	mt3i 141	. 2 |- ( Ord A -> ( A e. On \/ A = On ) )
11		eloni 5733	. . 3 |- ( A e. On -> Ord A )
12		ordeq 5730	. . . 4 |- ( A = On -> ( Ord A <-> Ord On ) )
13	4, 12	mpbiri 248	. . 3 |- ( A = On -> Ord A )
14	11, 13	jaoi 394	. 2 |- ( ( A e. On \/ A = On ) -> Ord A )
15	10, 14	impbii 199	1 |- ( Ord A <-> ( A e. On \/ A = On ) )

Syntax hints:   <-> wb 196   \/ wo 383   \/ w3o 1036   = wceq 1483   e. wcel 1990   _Vcvv 3200   Ord word 5722   Oncon0 5723

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-tr 4753  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-ord 5726  df-on 5727

This theorem is referenced by:  ordsson  6989  ssonprc  6992  ordunisuc  7032  orduninsuc  7043  limomss  7070  omon  7076  limom  7080  tfrlem14  7487  tfr2b  7492  unialeph  8924  ordtoplem  32434  ordcmp  32446