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Theorem ordeleqon 7721
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 ∈ On ∨ 𝐴 = On))

Proof of Theorem ordeleqon
StepHypRef Expression
1 onprc 7717 . . . 4 ¬ On ∈ V
2 elex 3466 . . . 4 (On ∈ 𝐴 → On ∈ V)
31, 2mto 196 . . 3 ¬ On ∈ 𝐴
4 ordon 7716 . . . . . 6 Ord On
5 ordtri3or 6354 . . . . . 6 ((Ord 𝐴 ∧ Ord On) → (𝐴 ∈ On ∨ 𝐴 = On ∨ On ∈ 𝐴))
64, 5mpan2 690 . . . . 5 (Ord 𝐴 → (𝐴 ∈ On ∨ 𝐴 = On ∨ On ∈ 𝐴))
7 df-3or 1089 . . . . 5 ((𝐴 ∈ On ∨ 𝐴 = On ∨ On ∈ 𝐴) ↔ ((𝐴 ∈ On ∨ 𝐴 = On) ∨ On ∈ 𝐴))
86, 7sylib 217 . . . 4 (Ord 𝐴 → ((𝐴 ∈ On ∨ 𝐴 = On) ∨ On ∈ 𝐴))
98ord 863 . . 3 (Ord 𝐴 → (¬ (𝐴 ∈ On ∨ 𝐴 = On) → On ∈ 𝐴))
103, 9mt3i 149 . 2 (Ord 𝐴 → (𝐴 ∈ On ∨ 𝐴 = On))
11 eloni 6332 . . 3 (𝐴 ∈ On → Ord 𝐴)
12 ordeq 6329 . . . 4 (𝐴 = On → (Ord 𝐴 ↔ Ord On))
134, 12mpbiri 258 . . 3 (𝐴 = On → Ord 𝐴)
1411, 13jaoi 856 . 2 ((𝐴 ∈ On ∨ 𝐴 = On) → Ord 𝐴)
1510, 14impbii 208 1 (Ord 𝐴 ↔ (𝐴 ∈ On ∨ 𝐴 = On))
Colors of variables: wff setvar class
Syntax hints:  wb 205  wo 846  w3o 1087   = wceq 1542  wcel 2107  Vcvv 3448  Ord word 6321  Oncon0 6322
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2708  ax-sep 5261  ax-nul 5268  ax-pr 5389
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2715  df-cleq 2729  df-clel 2815  df-ne 2945  df-ral 3066  df-rex 3075  df-rab 3411  df-v 3450  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-pss 3934  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-br 5111  df-opab 5173  df-tr 5228  df-eprel 5542  df-po 5550  df-so 5551  df-fr 5593  df-we 5595  df-ord 6325  df-on 6326
This theorem is referenced by:  ordsson  7722  ssonprc  7727  ordunisuc  7772  orduninsuc  7784  limomss  7812  omon  7819  limom  7823  tfrlem14  8342  tfr2b  8347  unialeph  10044  ordtoplem  34936  ordcmp  34948  onsupnmax  41591  dflim5  41693
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