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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 4518 | . 2 ⊢ (𝐴 ∈ ω → 𝐴 ∈ On) | |
2 | eloni 4292 | . 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 1480 Ord word 4279 Oncon0 4280 ωcom 4499 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-sep 4041 ax-nul 4049 ax-pow 4093 ax-pr 4126 ax-un 4350 ax-iinf 4497 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ral 2419 df-rex 2420 df-v 2683 df-dif 3068 df-un 3070 df-in 3072 df-ss 3079 df-nul 3359 df-pw 3507 df-sn 3528 df-pr 3529 df-uni 3732 df-int 3767 df-tr 4022 df-iord 4283 df-on 4285 df-suc 4288 df-iom 4500 |
This theorem is referenced by: nnsucsssuc 6381 nnsucuniel 6384 nntri1 6385 nnsseleq 6390 nntr2 6392 phplem1 6739 phplem2 6740 phplem3 6741 phplem4 6742 phplem4dom 6749 nndomo 6751 dif1en 6766 nnwetri 6797 unsnfi 6800 ctmlemr 6986 nnnninf 7016 piord 7112 addnidpig 7137 archnqq 7218 frecfzennn 10192 hashp1i 10549 ennnfonelemk 11902 ennnfonelemg 11905 ennnfonelemhf1o 11915 ennnfonelemhom 11917 ctinfom 11930 nnsf 13188 peano4nninf 13189 nninfalllemn 13191 nninfsellemeq 13199 |
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