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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 4128 | . 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 103 ∈ wcel 1433 Vcvv 2601 Ord word 4117 Oncon0 4118 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 |
This theorem depends on definitions: df-bi 115 df-tru 1287 df-nf 1390 df-sb 1686 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ral 2353 df-rex 2354 df-v 2603 df-in 2979 df-ss 2986 df-uni 3602 df-tr 3876 df-iord 4121 df-on 4123 |
This theorem is referenced by: tron 4137 0elon 4147 ordtriexmidlem 4263 ontr2exmid 4268 ordtri2or2exmidlem 4269 onsucelsucexmidlem 4272 bj-omelon 10756 |
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