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Mirrors > Home > ILE Home > Th. List > elon | GIF version |
Description: An ordinal number is an ordinal set. (Contributed by NM, 5-Jun-1994.) |
Ref | Expression |
---|---|
elon.1 | ⊢ 𝐴 ∈ V |
Ref | Expression |
---|---|
elon | ⊢ (𝐴 ∈ On ↔ Ord 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elon.1 | . 2 ⊢ 𝐴 ∈ V | |
2 | elong 4110 | . 2 ⊢ (𝐴 ∈ V → (𝐴 ∈ On ↔ Ord 𝐴)) | |
3 | 1, 2 | ax-mp 7 | 1 ⊢ (𝐴 ∈ On ↔ Ord 𝐴) |
Colors of variables: wff set class |
Syntax hints: ↔ wb 98 ∈ wcel 1393 Vcvv 2557 Ord word 4099 Oncon0 4100 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 |
This theorem depends on definitions: df-bi 110 df-tru 1246 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ral 2311 df-rex 2312 df-v 2559 df-in 2924 df-ss 2931 df-uni 3581 df-tr 3855 df-iord 4103 df-on 4105 |
This theorem is referenced by: tron 4119 0elon 4129 ordtriexmidlem 4245 ontr2exmid 4250 ordtri2or2exmidlem 4251 onsucelsucexmidlem 4254 bj-omelon 10086 |
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