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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 4349 | . 2 ⊢ ω ∈ On | |
2 | 1 | oneli 4183 | 1 ⊢ (𝐴 ∈ ω → 𝐴 ∈ On) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 1433 Oncon0 4118 ωcom 4331 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-13 1444 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-nul 3904 ax-pow 3948 ax-pr 3964 ax-un 4188 ax-iinf 4329 |
This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-nf 1390 df-sb 1686 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ral 2353 df-rex 2354 df-v 2603 df-dif 2975 df-un 2977 df-in 2979 df-ss 2986 df-nul 3252 df-pw 3384 df-sn 3404 df-pr 3405 df-uni 3602 df-int 3637 df-tr 3876 df-iord 4121 df-on 4123 df-suc 4126 df-iom 4332 |
This theorem is referenced by: nnoni 4351 nnord 4352 omsson 4353 frecsuclem1 6010 frecsuclemdm 6011 frecrdg 6015 onasuc 6069 onmsuc 6075 nna0 6076 nnm0 6077 nnasuc 6078 nnmsuc 6079 nnsucelsuc 6093 nnsucsssuc 6094 nntri2or2 6099 nnaordi 6104 nnaword1 6109 nnaordex 6123 phpelm 6352 phplem4on 6353 finnum 6452 pion 6500 prarloclemlo 6684 |
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