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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 3499 | . . 3 ⊢ (𝐴 ∈ On → 𝐴 ∈ V) | |
2 | elong 6394 | . . 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 2106 Vcvv 3478 Ord word 6385 Oncon0 6386 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1540 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-ral 3060 df-v 3480 df-ss 3980 df-uni 4913 df-tr 5266 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-ord 6389 df-on 6390 |
This theorem is referenced by: ordsuci 7828 onsucb 7837 tfrlem12 8428 tfrlem13 8429 gruina 10856 bdayimaon 27753 noeta2 27844 etasslt2 27874 oldlim 27940 oaltublim 43280 omord2lim 43290 oaun3lem3 43366 nadd2rabon 43377 nadd1rabon 43381 |
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