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Mirrors > Home > ILE Home > Th. List > nnord | GIF version |
Description: A natural number is ordinal. (Contributed by NM, 17-Oct-1995.) |
Ref | Expression |
---|---|
nnord | ⊢ (𝐴 ∈ ω → Ord 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nnon 4452 | . 2 ⊢ (𝐴 ∈ ω → 𝐴 ∈ On) | |
2 | eloni 4226 | . 2 ⊢ (𝐴 ∈ On → Ord 𝐴) | |
3 | 1, 2 | syl 14 | 1 ⊢ (𝐴 ∈ ω → Ord 𝐴) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 1445 Ord word 4213 Oncon0 4214 ωcom 4433 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 582 ax-in2 583 ax-io 668 ax-5 1388 ax-7 1389 ax-gen 1390 ax-ie1 1434 ax-ie2 1435 ax-8 1447 ax-10 1448 ax-11 1449 ax-i12 1450 ax-bndl 1451 ax-4 1452 ax-13 1456 ax-14 1457 ax-17 1471 ax-i9 1475 ax-ial 1479 ax-i5r 1480 ax-ext 2077 ax-sep 3978 ax-nul 3986 ax-pow 4030 ax-pr 4060 ax-un 4284 ax-iinf 4431 |
This theorem depends on definitions: df-bi 116 df-3an 929 df-tru 1299 df-nf 1402 df-sb 1700 df-clab 2082 df-cleq 2088 df-clel 2091 df-nfc 2224 df-ral 2375 df-rex 2376 df-v 2635 df-dif 3015 df-un 3017 df-in 3019 df-ss 3026 df-nul 3303 df-pw 3451 df-sn 3472 df-pr 3473 df-uni 3676 df-int 3711 df-tr 3959 df-iord 4217 df-on 4219 df-suc 4222 df-iom 4434 |
This theorem is referenced by: nnsucsssuc 6293 nnsucuniel 6296 nntri1 6297 nnsseleq 6302 phplem1 6648 phplem2 6649 phplem3 6650 phplem4 6651 phplem4dom 6658 nndomo 6660 dif1en 6675 nnwetri 6706 unsnfi 6709 ctmlemr 6870 nnnninf 6894 piord 6967 addnidpig 6992 archnqq 7073 frecfzennn 9982 hashp1i 10349 nnsf 12604 peano4nninf 12605 nninfalllemn 12607 nninfsellemeq 12615 |
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