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| Mirrors > Home > MPE Home > Th. List > dford5 | Structured version Visualization version GIF version | ||
| Description: A class is ordinal iff it is a subclass of On and transitive. (Contributed by Scott Fenton, 21-Nov-2021.) | 
| Ref | Expression | 
|---|---|
| dford5 | ⊢ (Ord 𝐴 ↔ (𝐴 ⊆ On ∧ Tr 𝐴)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | ordsson 7804 | . . 3 ⊢ (Ord 𝐴 → 𝐴 ⊆ On) | |
| 2 | ordtr 6397 | . . 3 ⊢ (Ord 𝐴 → Tr 𝐴) | |
| 3 | 1, 2 | jca 511 | . 2 ⊢ (Ord 𝐴 → (𝐴 ⊆ On ∧ Tr 𝐴)) | 
| 4 | epweon 7796 | . . . 4 ⊢ E We On | |
| 5 | wess 5670 | . . . 4 ⊢ (𝐴 ⊆ On → ( E We On → E We 𝐴)) | |
| 6 | 4, 5 | mpi 20 | . . 3 ⊢ (𝐴 ⊆ On → E We 𝐴) | 
| 7 | df-ord 6386 | . . . . 5 ⊢ (Ord 𝐴 ↔ (Tr 𝐴 ∧ E We 𝐴)) | |
| 8 | 7 | biimpri 228 | . . . 4 ⊢ ((Tr 𝐴 ∧ E We 𝐴) → Ord 𝐴) | 
| 9 | 8 | ancoms 458 | . . 3 ⊢ (( E We 𝐴 ∧ Tr 𝐴) → Ord 𝐴) | 
| 10 | 6, 9 | sylan 580 | . 2 ⊢ ((𝐴 ⊆ On ∧ Tr 𝐴) → Ord 𝐴) | 
| 11 | 3, 10 | impbii 209 | 1 ⊢ (Ord 𝐴 ↔ (𝐴 ⊆ On ∧ Tr 𝐴)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ↔ wb 206 ∧ wa 395 ⊆ wss 3950 Tr wtr 5258 E cep 5582 We wwe 5635 Ord word 6382 Oncon0 6383 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pr 5431 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-br 5143 df-opab 5205 df-tr 5259 df-eprel 5583 df-po 5591 df-so 5592 df-fr 5636 df-we 5638 df-ord 6386 df-on 6387 | 
| This theorem is referenced by: nosupno 27749 noinfno 27764 nadd2rabord 43403 nadd1rabord 43407 | 
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