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Mirrors > Home > ILE Home > Th. List > nnon | GIF version |
Description: A natural number is an ordinal number. (Contributed by NM, 27-Jun-1994.) |
Ref | Expression |
---|---|
nnon | ⊢ (𝐴 ∈ ω → 𝐴 ∈ On) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | omelon 4522 | . 2 ⊢ ω ∈ On | |
2 | 1 | oneli 4350 | 1 ⊢ (𝐴 ∈ ω → 𝐴 ∈ On) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 1480 Oncon0 4285 ωcom 4504 |
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 2121 ax-sep 4046 ax-nul 4054 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-iinf 4502 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-rex 2422 df-v 2688 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-nul 3364 df-pw 3512 df-sn 3533 df-pr 3534 df-uni 3737 df-int 3772 df-tr 4027 df-iord 4288 df-on 4290 df-suc 4293 df-iom 4505 |
This theorem is referenced by: nnoni 4524 nnord 4525 omsson 4526 nnsucpred 4530 nnpredcl 4536 frecrdg 6305 onasuc 6362 onmsuc 6369 nna0 6370 nnm0 6371 nnasuc 6372 nnmsuc 6373 nnsucelsuc 6387 nnsucsssuc 6388 nntri2or2 6394 nntr2 6399 nnaordi 6404 nnaword1 6409 nnaordex 6423 phpelm 6760 phplem4on 6761 omp1eomlem 6979 finnum 7039 pion 7118 prarloclemlo 7302 ennnfonelemk 11913 pwle2 13193 |
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