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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 7506 | . . 3 ⊢ (Ord 𝐴 → 𝐴 ⊆ On) | |
2 | ordtr 6207 | . . 3 ⊢ (Ord 𝐴 → Tr 𝐴) | |
3 | 1, 2 | jca 514 | . 2 ⊢ (Ord 𝐴 → (𝐴 ⊆ On ∧ Tr 𝐴)) |
4 | epweon 7499 | . . . 4 ⊢ E We On | |
5 | wess 5544 | . . . 4 ⊢ (𝐴 ⊆ On → ( E We On → E We 𝐴)) | |
6 | 4, 5 | mpi 20 | . . 3 ⊢ (𝐴 ⊆ On → E We 𝐴) |
7 | df-ord 6196 | . . . . 5 ⊢ (Ord 𝐴 ↔ (Tr 𝐴 ∧ E We 𝐴)) | |
8 | 7 | biimpri 230 | . . . 4 ⊢ ((Tr 𝐴 ∧ E We 𝐴) → Ord 𝐴) |
9 | 8 | ancoms 461 | . . 3 ⊢ (( E We 𝐴 ∧ Tr 𝐴) → Ord 𝐴) |
10 | 6, 9 | sylan 582 | . 2 ⊢ ((𝐴 ⊆ On ∧ Tr 𝐴) → Ord 𝐴) |
11 | 3, 10 | impbii 211 | 1 ⊢ (Ord 𝐴 ↔ (𝐴 ⊆ On ∧ Tr 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 ∧ wa 398 ⊆ wss 3938 Tr wtr 5174 E cep 5466 We wwe 5515 Ord word 6192 Oncon0 6193 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pr 5332 ax-un 7463 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-br 5069 df-opab 5131 df-tr 5175 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-ord 6196 df-on 6197 |
This theorem is referenced by: nosupno 33205 |
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