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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 6396 | . 2 ⊢ (𝐴 ∈ On → Ord 𝐴) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ Ord 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2106 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: onirri 6499 onsucssi 7862 ord1eln01 8533 ord2eln012 8534 oawordeulem 8591 omopthi 8698 en2 9313 en3 9314 ssttrcl 9753 ttrcltr 9754 dmttrcl 9759 ttrclselem2 9764 bndrank 9879 rankprb 9889 rankuniss 9904 rankelun 9910 rankelpr 9911 rankelop 9912 rankmapu 9916 rankxplim3 9919 rankxpsuc 9920 cardlim 10010 carduni 10019 dfac8b 10069 alephdom2 10125 alephfp 10146 dfac12lem2 10183 dju1p1e2ALT 10213 cfsmolem 10308 ttukeylem6 10552 ttukeylem7 10553 unsnen 10591 efgmnvl 19747 nogt01o 27756 scutbdaybnd2lim 27877 slerec 27879 bday1s 27891 cuteq1 27893 newbday 27955 negsproplem7 28081 mulsproplem13 28169 mulsproplem14 28170 sltonold 28298 pw2bday 28433 hfuni 36166 finxpsuclem 37380 pwfi2f1o 43085 |
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