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Theorem ordeleqon 7030
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 7026 . . . 4 ¬ On ∈ V
2 elex 3243 . . . 4 (On ∈ 𝐴 → On ∈ V)
31, 2mto 188 . . 3 ¬ On ∈ 𝐴
4 ordon 7024 . . . . . 6 Ord On
5 ordtri3or 5793 . . . . . 6 ((Ord 𝐴 ∧ Ord On) → (𝐴 ∈ On ∨ 𝐴 = On ∨ On ∈ 𝐴))
64, 5mpan2 707 . . . . 5 (Ord 𝐴 → (𝐴 ∈ On ∨ 𝐴 = On ∨ On ∈ 𝐴))
7 df-3or 1055 . . . . 5 ((𝐴 ∈ On ∨ 𝐴 = On ∨ On ∈ 𝐴) ↔ ((𝐴 ∈ On ∨ 𝐴 = On) ∨ On ∈ 𝐴))
86, 7sylib 208 . . . 4 (Ord 𝐴 → ((𝐴 ∈ On ∨ 𝐴 = On) ∨ On ∈ 𝐴))
98ord 391 . . 3 (Ord 𝐴 → (¬ (𝐴 ∈ On ∨ 𝐴 = On) → On ∈ 𝐴))
103, 9mt3i 141 . 2 (Ord 𝐴 → (𝐴 ∈ On ∨ 𝐴 = On))
11 eloni 5771 . . 3 (𝐴 ∈ On → Ord 𝐴)
12 ordeq 5768 . . . 4 (𝐴 = On → (Ord 𝐴 ↔ Ord On))
134, 12mpbiri 248 . . 3 (𝐴 = On → Ord 𝐴)
1411, 13jaoi 393 . 2 ((𝐴 ∈ On ∨ 𝐴 = On) → Ord 𝐴)
1510, 14impbii 199 1 (Ord 𝐴 ↔ (𝐴 ∈ On ∨ 𝐴 = On))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wo 382  w3o 1053   = wceq 1523  wcel 2030  Vcvv 3231  Ord word 5760  Oncon0 5761
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-tr 4786  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-ord 5764  df-on 5765
This theorem is referenced by:  ordsson  7031  ssonprc  7034  ordunisuc  7074  orduninsuc  7085  limomss  7112  omon  7118  limom  7122  tfrlem14  7532  tfr2b  7537  unialeph  8962  ordtoplem  32559  ordcmp  32571
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