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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 6093 | . 2 ⊢ 1𝑜 ∈ ω | |
2 | 1n0 6016 | . 2 ⊢ 1𝑜 ≠ ∅ | |
3 | elni 6406 | . 2 ⊢ (1𝑜 ∈ N ↔ (1𝑜 ∈ ω ∧ 1𝑜 ≠ ∅)) | |
4 | 1, 2, 3 | mpbir2an 849 | 1 ⊢ 1𝑜 ∈ N |
Colors of variables: wff set class |
Syntax hints: ∈ wcel 1393 ≠ wne 2204 ∅c0 3224 ωcom 4313 1𝑜c1o 5994 Ncnpi 6370 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-13 1404 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 ax-nul 3883 ax-pow 3927 ax-pr 3944 ax-un 4170 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ne 2206 df-ral 2311 df-rex 2312 df-v 2559 df-dif 2920 df-un 2922 df-in 2924 df-ss 2931 df-nul 3225 df-pw 3361 df-sn 3381 df-pr 3382 df-uni 3581 df-int 3616 df-suc 4108 df-iom 4314 df-1o 6001 df-ni 6402 |
This theorem is referenced by: mulidpi 6416 1lt2pi 6438 nlt1pig 6439 indpi 6440 1nq 6464 1qec 6486 mulidnq 6487 1lt2nq 6504 archnqq 6515 prarloclemarch 6516 prarloclemarch2 6517 nnnq 6520 ltnnnq 6521 nq0m0r 6554 nq0a0 6555 addpinq1 6562 nq02m 6563 prarloclemlt 6591 prarloclemlo 6592 prarloclemn 6597 prarloclemcalc 6600 nqprm 6640 caucvgprlemm 6766 caucvgprprlemml 6792 caucvgprprlemmu 6793 caucvgsrlemasr 6874 caucvgsr 6886 nntopi 6968 |
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