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| Description: Ordinal 3 is an ordinal class. (Contributed by BTernaryTau, 6-Jan-2025.) | 
| Ref | Expression | 
|---|---|
| ord3 | ⊢ Ord 3o | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | 2on 8520 | . . 3 ⊢ 2o ∈ On | |
| 2 | eloni 6394 | . . 3 ⊢ (2o ∈ On → Ord 2o) | |
| 3 | ordsuci 7828 | . . 3 ⊢ (Ord 2o → Ord suc 2o) | |
| 4 | 1, 2, 3 | mp2b 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)) | |
| 7 | 5, 6 | ax-mp 5 | . 2 ⊢ (Ord 3o ↔ Ord suc 2o) | 
| 8 | 4, 7 | mpbir 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 | 
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