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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 3509 | . . 3 ⊢ (𝐴 ∈ On → 𝐴 ∈ V) | |
2 | elong 6403 | . . 3 ⊢ (𝐴 ∈ V → (𝐴 ∈ On ↔ Ord 𝐴)) | |
3 | 1, 2 | biadanii 821 | . 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 3488 Ord word 6394 Oncon0 6395 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2711 |
This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1540 df-ex 1778 df-sb 2065 df-clab 2718 df-cleq 2732 df-clel 2819 df-ral 3068 df-v 3490 df-ss 3993 df-uni 4932 df-tr 5284 df-po 5607 df-so 5608 df-fr 5652 df-we 5654 df-ord 6398 df-on 6399 |
This theorem is referenced by: ordsuci 7844 onsucb 7853 tfrlem12 8445 tfrlem13 8446 gruina 10887 bdayimaon 27756 noeta2 27847 etasslt2 27877 oldlim 27943 oaltublim 43252 omord2lim 43262 oaun3lem3 43338 nadd2rabon 43349 nadd1rabon 43353 |
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