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Theorem ordeleqon 7313
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 7309 . . . 4 ¬ On ∈ V
2 elex 3427 . . . 4 (On ∈ 𝐴 → On ∈ V)
31, 2mto 189 . . 3 ¬ On ∈ 𝐴
4 ordon 7308 . . . . . 6 Ord On
5 ordtri3or 6055 . . . . . 6 ((Ord 𝐴 ∧ Ord On) → (𝐴 ∈ On ∨ 𝐴 = On ∨ On ∈ 𝐴))
64, 5mpan2 678 . . . . 5 (Ord 𝐴 → (𝐴 ∈ On ∨ 𝐴 = On ∨ On ∈ 𝐴))
7 df-3or 1069 . . . . 5 ((𝐴 ∈ On ∨ 𝐴 = On ∨ On ∈ 𝐴) ↔ ((𝐴 ∈ On ∨ 𝐴 = On) ∨ On ∈ 𝐴))
86, 7sylib 210 . . . 4 (Ord 𝐴 → ((𝐴 ∈ On ∨ 𝐴 = On) ∨ On ∈ 𝐴))
98ord 850 . . 3 (Ord 𝐴 → (¬ (𝐴 ∈ On ∨ 𝐴 = On) → On ∈ 𝐴))
103, 9mt3i 144 . 2 (Ord 𝐴 → (𝐴 ∈ On ∨ 𝐴 = On))
11 eloni 6033 . . 3 (𝐴 ∈ On → Ord 𝐴)
12 ordeq 6030 . . . 4 (𝐴 = On → (Ord 𝐴 ↔ Ord On))
134, 12mpbiri 250 . . 3 (𝐴 = On → Ord 𝐴)
1411, 13jaoi 843 . 2 ((𝐴 ∈ On ∨ 𝐴 = On) → Ord 𝐴)
1510, 14impbii 201 1 (Ord 𝐴 ↔ (𝐴 ∈ On ∨ 𝐴 = On))
Colors of variables: wff setvar class
Syntax hints:  wb 198  wo 833  w3o 1067   = wceq 1507  wcel 2050  Vcvv 3409  Ord word 6022  Oncon0 6023
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2744  ax-sep 5054  ax-nul 5061  ax-pr 5180  ax-un 7273
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3or 1069  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2584  df-clab 2753  df-cleq 2765  df-clel 2840  df-nfc 2912  df-ne 2962  df-ral 3087  df-rex 3088  df-rab 3091  df-v 3411  df-sbc 3676  df-dif 3826  df-un 3828  df-in 3830  df-ss 3837  df-pss 3839  df-nul 4173  df-if 4345  df-sn 4436  df-pr 4438  df-tp 4440  df-op 4442  df-uni 4707  df-br 4924  df-opab 4986  df-tr 5025  df-eprel 5311  df-po 5320  df-so 5321  df-fr 5360  df-we 5362  df-ord 6026  df-on 6027
This theorem is referenced by:  ordsson  7314  ssonprc  7317  ordunisuc  7357  orduninsuc  7368  limomss  7395  omon  7401  limom  7405  tfrlem14  7825  tfr2b  7830  unialeph  9315  ordtoplem  33303  ordcmp  33315
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