Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > 1nn | Unicode version |
Description: Peano postulate: 1 is a positive integer. (Contributed by NM, 11-Jan-1997.) |
Ref | Expression |
---|---|
1nn |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfnn2 8686 | . . . 4 | |
2 | 1 | eleq2i 2184 | . . 3 |
3 | 1re 7733 | . . . 4 | |
4 | elintg 3749 | . . . 4 | |
5 | 3, 4 | ax-mp 5 | . . 3 |
6 | 2, 5 | bitri 183 | . 2 |
7 | vex 2663 | . . . 4 | |
8 | eleq2 2181 | . . . . 5 | |
9 | eleq2 2181 | . . . . . 6 | |
10 | 9 | raleqbi1dv 2611 | . . . . 5 |
11 | 8, 10 | anbi12d 464 | . . . 4 |
12 | 7, 11 | elab 2802 | . . 3 |
13 | 12 | simplbi 272 | . 2 |
14 | 6, 13 | mprgbir 2467 | 1 |
Colors of variables: wff set class |
Syntax hints: wa 103 wb 104 wcel 1465 cab 2103 wral 2393 cint 3741 (class class class)co 5742 cr 7587 c1 7589 caddc 7591 cn 8684 |
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-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-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-1re 7682 |
This theorem depends on definitions: df-bi 116 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-v 2662 df-int 3742 df-inn 8685 |
This theorem is referenced by: nnind 8700 nn1suc 8703 2nn 8839 1nn0 8951 nn0p1nn 8974 1z 9038 neg1z 9044 elz2 9080 nneoor 9111 9p1e10 9142 indstr 9344 elnn1uz2 9357 zq 9374 qreccl 9390 fz01or 9846 exp3vallem 10249 exp1 10254 nnexpcl 10261 expnbnd 10370 3dec 10416 fac1 10430 faccl 10436 faclbnd3 10444 resqrexlemf1 10735 resqrexlemcalc3 10743 resqrexlemnmsq 10744 resqrexlemnm 10745 resqrexlemcvg 10746 resqrexlemglsq 10749 resqrexlemga 10750 sumsnf 11133 cvgratnnlemnexp 11248 cvgratnnlemfm 11253 cvgratnnlemrate 11254 cvgratnn 11255 eftlub 11310 eirraplem 11395 n2dvds1 11521 ndvdsp1 11541 gcd1 11587 bezoutr1 11633 ncoprmgcdne1b 11682 1nprm 11707 1idssfct 11708 isprm2lem 11709 qden1elz 11794 phicl2 11801 phi1 11806 phiprm 11810 exmidunben 11850 base0 11919 baseval 11922 baseid 11923 basendx 11924 basendxnn 11925 1strstrg 11968 2strstrg 11970 basendxnplusgndx 11976 basendxnmulrndx 11984 rngstrg 11985 lmodstrd 12003 topgrpstrd 12021 setsmsdsg 12560 trilpolemgt1 13128 |
Copyright terms: Public domain | W3C validator |