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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 6544 | . 2 ⊢ 1o ∈ ω | |
2 | 1n0 6456 | . 2 ⊢ 1o ≠ ∅ | |
3 | elni 7336 | . 2 ⊢ (1o ∈ N ↔ (1o ∈ ω ∧ 1o ≠ ∅)) | |
4 | 1, 2, 3 | mpbir2an 944 | 1 ⊢ 1o ∈ N |
Colors of variables: wff set class |
Syntax hints: ∈ wcel 2160 ≠ wne 2360 ∅c0 3437 ωcom 4607 1oc1o 6433 Ncnpi 7300 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-sep 4136 ax-nul 4144 ax-pow 4192 ax-pr 4227 ax-un 4451 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-nf 1472 df-sb 1774 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-ral 2473 df-rex 2474 df-v 2754 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-nul 3438 df-pw 3592 df-sn 3613 df-pr 3614 df-uni 3825 df-int 3860 df-suc 4389 df-iom 4608 df-1o 6440 df-ni 7332 |
This theorem is referenced by: mulidpi 7346 1lt2pi 7368 nlt1pig 7369 indpi 7370 1nq 7394 1qec 7416 mulidnq 7417 1lt2nq 7434 archnqq 7445 prarloclemarch 7446 prarloclemarch2 7447 nnnq 7450 ltnnnq 7451 nq0m0r 7484 nq0a0 7485 addpinq1 7492 nq02m 7493 prarloclemlt 7521 prarloclemlo 7522 prarloclemn 7527 prarloclemcalc 7530 nqprm 7570 caucvgprlemm 7696 caucvgprprlemml 7722 caucvgprprlemmu 7723 caucvgsrlemasr 7818 caucvgsr 7830 nntopi 7922 |
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