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Theorem ordeleqon 7487
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 7483 . . . 4 ¬ On ∈ V
2 elex 3462 . . . 4 (On ∈ 𝐴 → On ∈ V)
31, 2mto 200 . . 3 ¬ On ∈ 𝐴
4 ordon 7482 . . . . . 6 Ord On
5 ordtri3or 6195 . . . . . 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 1085 . . . . 5 ((𝐴 ∈ On ∨ 𝐴 = On ∨ On ∈ 𝐴) ↔ ((𝐴 ∈ On ∨ 𝐴 = On) ∨ On ∈ 𝐴))
86, 7sylib 221 . . . 4 (Ord 𝐴 → ((𝐴 ∈ On ∨ 𝐴 = On) ∨ On ∈ 𝐴))
98ord 861 . . 3 (Ord 𝐴 → (¬ (𝐴 ∈ On ∨ 𝐴 = On) → On ∈ 𝐴))
103, 9mt3i 151 . 2 (Ord 𝐴 → (𝐴 ∈ On ∨ 𝐴 = On))
11 eloni 6173 . . 3 (𝐴 ∈ On → Ord 𝐴)
12 ordeq 6170 . . . 4 (𝐴 = On → (Ord 𝐴 ↔ Ord On))
134, 12mpbiri 261 . . 3 (𝐴 = On → Ord 𝐴)
1411, 13jaoi 854 . 2 ((𝐴 ∈ On ∨ 𝐴 = On) → Ord 𝐴)
1510, 14impbii 212 1 (Ord 𝐴 ↔ (𝐴 ∈ On ∨ 𝐴 = On))
Colors of variables: wff setvar class
Syntax hints:  wb 209  wo 844  w3o 1083   = wceq 1538  wcel 2112  Vcvv 3444  Ord word 6162  Oncon0 6163
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pr 5298  ax-un 7445
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-rab 3118  df-v 3446  df-sbc 3724  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-pss 3903  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-tp 4533  df-op 4535  df-uni 4804  df-br 5034  df-opab 5096  df-tr 5140  df-eprel 5433  df-po 5442  df-so 5443  df-fr 5482  df-we 5484  df-ord 6166  df-on 6167
This theorem is referenced by:  ordsson  7488  ssonprc  7491  ordunisuc  7531  orduninsuc  7542  limomss  7569  omon  7575  limom  7579  tfrlem14  8014  tfr2b  8019  unialeph  9516  ordtoplem  33897  ordcmp  33909
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