MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ordeleqon Structured version   Visualization version   GIF version

Theorem ordeleqon 6880
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 6876 . . . 4 ¬ On ∈ V
2 elex 3185 . . . 4 (On ∈ 𝐴 → On ∈ V)
31, 2mto 187 . . 3 ¬ On ∈ 𝐴
4 ordon 6874 . . . . . 6 Ord On
5 ordtri3or 5672 . . . . . 6 ((Ord 𝐴 ∧ Ord On) → (𝐴 ∈ On ∨ 𝐴 = On ∨ On ∈ 𝐴))
64, 5mpan2 703 . . . . 5 (Ord 𝐴 → (𝐴 ∈ On ∨ 𝐴 = On ∨ On ∈ 𝐴))
7 df-3or 1032 . . . . 5 ((𝐴 ∈ On ∨ 𝐴 = On ∨ On ∈ 𝐴) ↔ ((𝐴 ∈ On ∨ 𝐴 = On) ∨ On ∈ 𝐴))
86, 7sylib 207 . . . 4 (Ord 𝐴 → ((𝐴 ∈ On ∨ 𝐴 = On) ∨ On ∈ 𝐴))
98ord 391 . . 3 (Ord 𝐴 → (¬ (𝐴 ∈ On ∨ 𝐴 = On) → On ∈ 𝐴))
103, 9mt3i 140 . 2 (Ord 𝐴 → (𝐴 ∈ On ∨ 𝐴 = On))
11 eloni 5650 . . 3 (𝐴 ∈ On → Ord 𝐴)
12 ordeq 5647 . . . 4 (𝐴 = On → (Ord 𝐴 ↔ Ord On))
134, 12mpbiri 247 . . 3 (𝐴 = On → Ord 𝐴)
1411, 13jaoi 393 . 2 ((𝐴 ∈ On ∨ 𝐴 = On) → Ord 𝐴)
1510, 14impbii 198 1 (Ord 𝐴 ↔ (𝐴 ∈ On ∨ 𝐴 = On))
Colors of variables: wff setvar class
Syntax hints:  wb 195  wo 382  w3o 1030   = wceq 1475  wcel 1977  Vcvv 3173  Ord word 5639  Oncon0 5640
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833  ax-un 6847
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-tr 4681  df-eprel 4949  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-ord 5643  df-on 5644
This theorem is referenced by:  ordsson  6881  ssonprc  6884  ordunisuc  6924  orduninsuc  6935  limomss  6962  omon  6968  limom  6972  tfrlem14  7374  tfr2b  7379  unialeph  8807  ordtoplem  31604  ordcmp  31616
  Copyright terms: Public domain W3C validator