MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ord3 Structured version   Visualization version   GIF version

Theorem ord3 8523
Description: Ordinal 3 is an ordinal class. (Contributed by BTernaryTau, 6-Jan-2025.)
Assertion
Ref Expression
ord3 Ord 3o

Proof of Theorem ord3
StepHypRef Expression
1 2on 8520 . . 3 2o ∈ On
2 eloni 6394 . . 3 (2o ∈ On → Ord 2o)
3 ordsuci 7828 . . 3 (Ord 2o → Ord suc 2o)
41, 2, 3mp2b 10 . 2 Ord suc 2o
5 df-3o 8508 . . 3 3o = suc 2o
6 ordeq 6391 . . 3 (3o = suc 2o → (Ord 3o ↔ Ord suc 2o))
75, 6ax-mp 5 . 2 (Ord 3o ↔ Ord suc 2o)
84, 7mpbir 231 1 Ord 3o
Colors of variables: wff setvar class
Syntax hints:  wb 206   = wceq 1540  wcel 2108  Ord word 6383  Oncon0 6384  suc csuc 6386  2oc2o 8500  3oc3o 8501
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-tr 5260  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-ord 6387  df-on 6388  df-suc 6390  df-1o 8506  df-2o 8507  df-3o 8508
This theorem is referenced by:  en4  9317
  Copyright terms: Public domain W3C validator