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Mirrors > Home > MPE Home > Th. List > ord3 | Structured version Visualization version GIF version |
Description: Ordinal 3 is an ordinal class. (Contributed by BTernaryTau, 6-Jan-2025.) |
Ref | Expression |
---|---|
ord3 | ⊢ Ord 3o |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 2on 8519 | . . 3 ⊢ 2o ∈ On | |
2 | eloni 6396 | . . 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 8507 | . . 3 ⊢ 3o = suc 2o | |
6 | ordeq 6393 | . . 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 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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