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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 3493 | . . 3 ⊢ (𝐴 ∈ On → 𝐴 ∈ V) | |
2 | elong 6364 | . . 3 ⊢ (𝐴 ∈ V → (𝐴 ∈ On ↔ Ord 𝐴)) | |
3 | 1, 2 | biadanii 821 | . 2 ⊢ (𝐴 ∈ On ↔ (𝐴 ∈ V ∧ Ord 𝐴)) |
4 | 3 | biancomi 464 | 1 ⊢ (𝐴 ∈ On ↔ (Ord 𝐴 ∧ 𝐴 ∈ V)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 397 ∈ wcel 2107 Vcvv 3475 Ord word 6355 Oncon0 6356 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 |
This theorem depends on definitions: df-bi 206 df-an 398 df-tru 1545 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-ral 3063 df-v 3477 df-in 3953 df-ss 3963 df-uni 4905 df-tr 5262 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-ord 6359 df-on 6360 |
This theorem is referenced by: ordsuci 7783 onsucb 7792 tfrlem12 8376 tfrlem13 8377 gruina 10800 bdayimaon 27163 noeta2 27253 etasslt2 27282 oldlim 27348 oaltublim 41911 omord2lim 41921 oaun3lem3 41997 nadd2rabon 42008 nadd1rabon 42012 |
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