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Mirrors > Home > ILE Home > Th. List > elon2 | GIF version |
Description: An ordinal number is an ordinal set. (Contributed by NM, 8-Feb-2004.) |
Ref | Expression |
---|---|
elon2 | ⊢ (𝐴 ∈ On ↔ (Ord 𝐴 ∧ 𝐴 ∈ V)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eloni 4267 | . . 3 ⊢ (𝐴 ∈ On → Ord 𝐴) | |
2 | elex 2671 | . . 3 ⊢ (𝐴 ∈ On → 𝐴 ∈ V) | |
3 | 1, 2 | jca 304 | . 2 ⊢ (𝐴 ∈ On → (Ord 𝐴 ∧ 𝐴 ∈ V)) |
4 | elong 4265 | . . 3 ⊢ (𝐴 ∈ V → (𝐴 ∈ On ↔ Ord 𝐴)) | |
5 | 4 | biimparc 297 | . 2 ⊢ ((Ord 𝐴 ∧ 𝐴 ∈ V) → 𝐴 ∈ On) |
6 | 3, 5 | impbii 125 | 1 ⊢ (𝐴 ∈ On ↔ (Ord 𝐴 ∧ 𝐴 ∈ V)) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 103 ↔ wb 104 ∈ wcel 1465 Vcvv 2660 Ord word 4254 Oncon0 4255 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 |
This theorem depends on definitions: df-bi 116 df-tru 1319 df-nf 1422 df-sb 1721 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ral 2398 df-rex 2399 df-v 2662 df-in 3047 df-ss 3054 df-uni 3707 df-tr 3997 df-iord 4258 df-on 4260 |
This theorem is referenced by: tfrexlem 6199 |
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