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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 7496 | . . 3 ⊢ (Ord 𝐴 → 𝐴 ⊆ On) | |
2 | ordtr 6176 | . . 3 ⊢ (Ord 𝐴 → Tr 𝐴) | |
3 | 1, 2 | jca 516 | . 2 ⊢ (Ord 𝐴 → (𝐴 ⊆ On ∧ Tr 𝐴)) |
4 | epweon 7489 | . . . 4 ⊢ E We On | |
5 | wess 5504 | . . . 4 ⊢ (𝐴 ⊆ On → ( E We On → E We 𝐴)) | |
6 | 4, 5 | mpi 20 | . . 3 ⊢ (𝐴 ⊆ On → E We 𝐴) |
7 | df-ord 6165 | . . . . 5 ⊢ (Ord 𝐴 ↔ (Tr 𝐴 ∧ E We 𝐴)) | |
8 | 7 | biimpri 231 | . . . 4 ⊢ ((Tr 𝐴 ∧ E We 𝐴) → Ord 𝐴) |
9 | 8 | ancoms 463 | . . 3 ⊢ (( E We 𝐴 ∧ Tr 𝐴) → Ord 𝐴) |
10 | 6, 9 | sylan 584 | . 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 400 ⊆ wss 3854 Tr wtr 5131 E cep 5427 We wwe 5475 Ord word 6161 Oncon0 6162 |
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 1912 ax-6 1971 ax-7 2016 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2159 ax-12 2176 ax-ext 2730 ax-sep 5162 ax-nul 5169 ax-pr 5291 ax-un 7452 |
This theorem depends on definitions: df-bi 210 df-an 401 df-or 846 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2071 df-clab 2737 df-cleq 2751 df-clel 2831 df-nfc 2899 df-ne 2950 df-ral 3073 df-rex 3074 df-rab 3077 df-v 3409 df-sbc 3694 df-dif 3857 df-un 3859 df-in 3861 df-ss 3871 df-pss 3873 df-nul 4222 df-if 4414 df-sn 4516 df-pr 4518 df-tp 4520 df-op 4522 df-uni 4792 df-br 5026 df-opab 5088 df-tr 5132 df-eprel 5428 df-po 5436 df-so 5437 df-fr 5476 df-we 5478 df-ord 6165 df-on 6166 |
This theorem is referenced by: nosupno 33456 noinfno 33471 |
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