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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 4492 | . 2 ⊢ ω ∈ On | |
2 | 1 | oneli 4320 | 1 ⊢ (𝐴 ∈ ω → 𝐴 ∈ On) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 1465 Oncon0 4255 ωcom 4474 |
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 588 ax-in2 589 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-13 1476 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-nul 4024 ax-pow 4068 ax-pr 4101 ax-un 4325 ax-iinf 4472 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 df-nf 1422 df-sb 1721 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ral 2398 df-rex 2399 df-v 2662 df-dif 3043 df-un 3045 df-in 3047 df-ss 3054 df-nul 3334 df-pw 3482 df-sn 3503 df-pr 3504 df-uni 3707 df-int 3742 df-tr 3997 df-iord 4258 df-on 4260 df-suc 4263 df-iom 4475 |
This theorem is referenced by: nnoni 4494 nnord 4495 omsson 4496 nnsucpred 4500 nnpredcl 4506 frecrdg 6273 onasuc 6330 onmsuc 6337 nna0 6338 nnm0 6339 nnasuc 6340 nnmsuc 6341 nnsucelsuc 6355 nnsucsssuc 6356 nntri2or2 6362 nntr2 6367 nnaordi 6372 nnaword1 6377 nnaordex 6391 phpelm 6728 phplem4on 6729 omp1eomlem 6947 finnum 7007 pion 7086 prarloclemlo 7270 ennnfonelemk 11840 pwle2 13120 |
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