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Mirrors > Home > MPE Home > Th. List > elon | Structured version Visualization version GIF version |
Description: An ordinal number is an ordinal set. (Contributed by NM, 5-Jun-1994.) |
Ref | Expression |
---|---|
elon.1 | ⊢ 𝐴 ∈ V |
Ref | Expression |
---|---|
elon | ⊢ (𝐴 ∈ On ↔ Ord 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elon.1 | . 2 ⊢ 𝐴 ∈ V | |
2 | elong 6382 | . 2 ⊢ (𝐴 ∈ V → (𝐴 ∈ On ↔ Ord 𝐴)) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ (𝐴 ∈ On ↔ Ord 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∈ wcel 2098 Vcvv 3473 Ord word 6373 Oncon0 6374 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2699 |
This theorem depends on definitions: df-bi 206 df-an 395 df-tru 1536 df-ex 1774 df-sb 2060 df-clab 2706 df-cleq 2720 df-clel 2806 df-ral 3059 df-v 3475 df-in 3956 df-ss 3966 df-uni 4913 df-tr 5270 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-ord 6377 df-on 6378 |
This theorem is referenced by: tron 6397 0elon 6428 smogt 8394 dfrecs3 8399 dfrecs3OLD 8400 rdglim2 8459 omeulem1 8609 naddcllem 8703 isfinite2 9332 r0weon 10043 cflim3 10293 inar1 10806 addsproplem7 27912 ellimits 35539 dford3lem2 42479 |
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