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Theorem ord3 8522
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 8519 . . 3 2o ∈ On
2 eloni 6396 . . 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 8507 . . 3 3o = suc 2o
6 ordeq 6393 . . 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 1537  wcel 2106  Ord word 6385  Oncon0 6386  suc csuc 6388  2oc2o 8499  3oc3o 8500
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-tr 5266  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-ord 6389  df-on 6390  df-suc 6392  df-1o 8505  df-2o 8506  df-3o 8507
This theorem is referenced by:  en4  9315
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