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Mirrors > Home > ILE Home > Th. List > 1pi | GIF version |
Description: Ordinal 'one' is a positive integer. (Contributed by NM, 29-Oct-1995.) |
Ref | Expression |
---|---|
1pi | ⊢ 1𝑜 ∈ N |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1onn 6279 | . 2 ⊢ 1𝑜 ∈ ω | |
2 | 1n0 6197 | . 2 ⊢ 1𝑜 ≠ ∅ | |
3 | elni 6867 | . 2 ⊢ (1𝑜 ∈ N ↔ (1𝑜 ∈ ω ∧ 1𝑜 ≠ ∅)) | |
4 | 1, 2, 3 | mpbir2an 888 | 1 ⊢ 1𝑜 ∈ N |
Colors of variables: wff set class |
Syntax hints: ∈ wcel 1438 ≠ wne 2255 ∅c0 3286 ωcom 4405 1𝑜c1o 6174 Ncnpi 6831 |
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 579 ax-in2 580 ax-io 665 ax-5 1381 ax-7 1382 ax-gen 1383 ax-ie1 1427 ax-ie2 1428 ax-8 1440 ax-10 1441 ax-11 1442 ax-i12 1443 ax-bndl 1444 ax-4 1445 ax-13 1449 ax-14 1450 ax-17 1464 ax-i9 1468 ax-ial 1472 ax-i5r 1473 ax-ext 2070 ax-sep 3957 ax-nul 3965 ax-pow 4009 ax-pr 4036 ax-un 4260 |
This theorem depends on definitions: df-bi 115 df-3an 926 df-tru 1292 df-nf 1395 df-sb 1693 df-clab 2075 df-cleq 2081 df-clel 2084 df-nfc 2217 df-ne 2256 df-ral 2364 df-rex 2365 df-v 2621 df-dif 3001 df-un 3003 df-in 3005 df-ss 3012 df-nul 3287 df-pw 3431 df-sn 3452 df-pr 3453 df-uni 3654 df-int 3689 df-suc 4198 df-iom 4406 df-1o 6181 df-ni 6863 |
This theorem is referenced by: mulidpi 6877 1lt2pi 6899 nlt1pig 6900 indpi 6901 1nq 6925 1qec 6947 mulidnq 6948 1lt2nq 6965 archnqq 6976 prarloclemarch 6977 prarloclemarch2 6978 nnnq 6981 ltnnnq 6982 nq0m0r 7015 nq0a0 7016 addpinq1 7023 nq02m 7024 prarloclemlt 7052 prarloclemlo 7053 prarloclemn 7058 prarloclemcalc 7061 nqprm 7101 caucvgprlemm 7227 caucvgprprlemml 7253 caucvgprprlemmu 7254 caucvgsrlemasr 7335 caucvgsr 7347 nntopi 7429 |
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