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| Mirrors > Home > MPE Home > Th. List > onordi | Structured version Visualization version GIF version | ||
| Description: An ordinal number is an ordinal class. (Contributed by NM, 11-Jun-1994.) | 
| Ref | Expression | 
|---|---|
| on.1 | ⊢ 𝐴 ∈ On | 
| Ref | Expression | 
|---|---|
| onordi | ⊢ Ord 𝐴 | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | on.1 | . 2 ⊢ 𝐴 ∈ On | |
| 2 | eloni 6393 | . 2 ⊢ (𝐴 ∈ On → Ord 𝐴) | |
| 3 | 1, 2 | ax-mp 5 | 1 ⊢ Ord 𝐴 | 
| Colors of variables: wff setvar class | 
| Syntax hints: ∈ wcel 2107 Ord word 6382 Oncon0 6383 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-ext 2707 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1542 df-ex 1779 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-ral 3061 df-v 3481 df-ss 3967 df-uni 4907 df-tr 5259 df-po 5591 df-so 5592 df-fr 5636 df-we 5638 df-ord 6386 df-on 6387 | 
| This theorem is referenced by: onirri 6496 onsucssi 7863 ord1eln01 8535 ord2eln012 8536 oawordeulem 8593 omopthi 8700 en2 9316 en3 9317 ssttrcl 9756 ttrcltr 9757 dmttrcl 9762 ttrclselem2 9767 bndrank 9882 rankprb 9892 rankuniss 9907 rankelun 9913 rankelpr 9914 rankelop 9915 rankmapu 9919 rankxplim3 9922 rankxpsuc 9923 cardlim 10013 carduni 10022 dfac8b 10072 alephdom2 10128 alephfp 10149 dfac12lem2 10186 dju1p1e2ALT 10216 cfsmolem 10311 ttukeylem6 10555 ttukeylem7 10556 unsnen 10594 efgmnvl 19733 nogt01o 27742 scutbdaybnd2lim 27863 slerec 27865 bday1s 27877 cuteq1 27879 newbday 27941 negsproplem7 28067 mulsproplem13 28155 mulsproplem14 28156 sltonold 28284 pw2bday 28419 hfuni 36186 finxpsuclem 37399 pwfi2f1o 43113 | 
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