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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 8462 | . . 3 ⊢ 2o ∈ On | |
2 | eloni 6363 | . . 3 ⊢ (2o ∈ On → Ord 2o) | |
3 | ordsuci 7779 | . . 3 ⊢ (Ord 2o → Ord suc 2o) | |
4 | 1, 2, 3 | mp2b 10 | . 2 ⊢ Ord suc 2o |
5 | df-3o 8450 | . . 3 ⊢ 3o = suc 2o | |
6 | ordeq 6360 | . . 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 230 | 1 ⊢ Ord 3o |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 = wceq 1541 ∈ wcel 2106 Ord word 6352 Oncon0 6353 suc csuc 6355 2oc2o 8442 3oc3o 8443 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2702 ax-sep 5292 ax-nul 5299 ax-pr 5420 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-sb 2068 df-clab 2709 df-cleq 2723 df-clel 2809 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3432 df-v 3475 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4523 df-pw 4598 df-sn 4623 df-pr 4625 df-op 4629 df-uni 4902 df-br 5142 df-opab 5204 df-tr 5259 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-ord 6356 df-on 6357 df-suc 6359 df-1o 8448 df-2o 8449 df-3o 8450 |
This theorem is referenced by: en4 9266 |
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