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Mirrors > Home > MPE Home > Th. List > Mathboxes > 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 7567 | . . 3 ⊢ (Ord 𝐴 → 𝐴 ⊆ On) | |
2 | ordtr 6227 | . . 3 ⊢ (Ord 𝐴 → Tr 𝐴) | |
3 | 1, 2 | jca 515 | . 2 ⊢ (Ord 𝐴 → (𝐴 ⊆ On ∧ Tr 𝐴)) |
4 | epweon 7560 | . . . 4 ⊢ E We On | |
5 | wess 5538 | . . . 4 ⊢ (𝐴 ⊆ On → ( E We On → E We 𝐴)) | |
6 | 4, 5 | mpi 20 | . . 3 ⊢ (𝐴 ⊆ On → E We 𝐴) |
7 | df-ord 6216 | . . . . 5 ⊢ (Ord 𝐴 ↔ (Tr 𝐴 ∧ E We 𝐴)) | |
8 | 7 | biimpri 231 | . . . 4 ⊢ ((Tr 𝐴 ∧ E We 𝐴) → Ord 𝐴) |
9 | 8 | ancoms 462 | . . 3 ⊢ (( E We 𝐴 ∧ Tr 𝐴) → Ord 𝐴) |
10 | 6, 9 | sylan 583 | . 2 ⊢ ((𝐴 ⊆ On ∧ Tr 𝐴) → Ord 𝐴) |
11 | 3, 10 | impbii 212 | 1 ⊢ (Ord 𝐴 ↔ (𝐴 ⊆ On ∧ Tr 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 ∧ wa 399 ⊆ wss 3866 Tr wtr 5161 E cep 5459 We wwe 5508 Ord word 6212 Oncon0 6213 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-11 2158 ax-ext 2708 ax-sep 5192 ax-nul 5199 ax-pr 5322 ax-un 7523 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-sb 2071 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-ral 3066 df-rex 3067 df-rab 3070 df-v 3410 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-pss 3885 df-nul 4238 df-if 4440 df-sn 4542 df-pr 4544 df-tp 4546 df-op 4548 df-uni 4820 df-br 5054 df-opab 5116 df-tr 5162 df-eprel 5460 df-po 5468 df-so 5469 df-fr 5509 df-we 5511 df-ord 6216 df-on 6217 |
This theorem is referenced by: nosupno 33643 noinfno 33658 |
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