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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 7722 | . . 3 ⊢ (Ord 𝐴 → 𝐴 ⊆ On) | |
2 | ordtr 6336 | . . 3 ⊢ (Ord 𝐴 → Tr 𝐴) | |
3 | 1, 2 | jca 513 | . 2 ⊢ (Ord 𝐴 → (𝐴 ⊆ On ∧ Tr 𝐴)) |
4 | epweon 7714 | . . . 4 ⊢ E We On | |
5 | wess 5625 | . . . 4 ⊢ (𝐴 ⊆ On → ( E We On → E We 𝐴)) | |
6 | 4, 5 | mpi 20 | . . 3 ⊢ (𝐴 ⊆ On → E We 𝐴) |
7 | df-ord 6325 | . . . . 5 ⊢ (Ord 𝐴 ↔ (Tr 𝐴 ∧ E We 𝐴)) | |
8 | 7 | biimpri 227 | . . . 4 ⊢ ((Tr 𝐴 ∧ E We 𝐴) → Ord 𝐴) |
9 | 8 | ancoms 460 | . . 3 ⊢ (( E We 𝐴 ∧ Tr 𝐴) → Ord 𝐴) |
10 | 6, 9 | sylan 581 | . 2 ⊢ ((𝐴 ⊆ On ∧ Tr 𝐴) → Ord 𝐴) |
11 | 3, 10 | impbii 208 | 1 ⊢ (Ord 𝐴 ↔ (𝐴 ⊆ On ∧ Tr 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 397 ⊆ wss 3915 Tr wtr 5227 E cep 5541 We wwe 5592 Ord word 6321 Oncon0 6322 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2708 ax-sep 5261 ax-nul 5268 ax-pr 5389 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2715 df-cleq 2729 df-clel 2815 df-ne 2945 df-ral 3066 df-rex 3075 df-rab 3411 df-v 3450 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-pss 3934 df-nul 4288 df-if 4492 df-pw 4567 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-br 5111 df-opab 5173 df-tr 5228 df-eprel 5542 df-po 5550 df-so 5551 df-fr 5593 df-we 5595 df-ord 6325 df-on 6326 |
This theorem is referenced by: nosupno 27067 noinfno 27082 nadd2rabord 41730 nadd1rabord 41734 |
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