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| Mirrors > Home > MPE Home > Th. List > elon2 | Structured version Visualization version GIF version | ||
| Description: An ordinal number is an ordinal set. Part of Definition 1.2 of [Schloeder] p. 1. (Contributed by NM, 8-Feb-2004.) |
| Ref | Expression |
|---|---|
| elon2 | ⊢ (𝐴 ∈ On ↔ (Ord 𝐴 ∧ 𝐴 ∈ V)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | elex 3501 | . . 3 ⊢ (𝐴 ∈ On → 𝐴 ∈ V) | |
| 2 | elong 6392 | . . 3 ⊢ (𝐴 ∈ V → (𝐴 ∈ On ↔ Ord 𝐴)) | |
| 3 | 1, 2 | biadanii 822 | . 2 ⊢ (𝐴 ∈ On ↔ (𝐴 ∈ V ∧ Ord 𝐴)) |
| 4 | 3 | biancomi 462 | 1 ⊢ (𝐴 ∈ On ↔ (Ord 𝐴 ∧ 𝐴 ∈ V)) |
| Colors of variables: wff setvar class |
| Syntax hints: ↔ wb 206 ∧ wa 395 ∈ wcel 2108 Vcvv 3480 Ord word 6383 Oncon0 6384 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1543 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-v 3482 df-ss 3968 df-uni 4908 df-tr 5260 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-ord 6387 df-on 6388 |
| This theorem is referenced by: ordsuci 7828 onsucb 7837 tfrlem12 8429 tfrlem13 8430 gruina 10858 bdayimaon 27738 noeta2 27829 etasslt2 27859 oldlim 27925 oaltublim 43303 omord2lim 43313 oaun3lem3 43389 nadd2rabon 43400 nadd1rabon 43404 |
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