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Mirrors > Home > ILE Home > Th. List > 2nn | GIF version |
Description: 2 is a positive integer. (Contributed by NM, 20-Aug-2001.) |
Ref | Expression |
---|---|
2nn | ⊢ 2 ∈ ℕ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-2 8747 | . 2 ⊢ 2 = (1 + 1) | |
2 | 1nn 8699 | . . 3 ⊢ 1 ∈ ℕ | |
3 | peano2nn 8700 | . . 3 ⊢ (1 ∈ ℕ → (1 + 1) ∈ ℕ) | |
4 | 2, 3 | ax-mp 5 | . 2 ⊢ (1 + 1) ∈ ℕ |
5 | 1, 4 | eqeltri 2190 | 1 ⊢ 2 ∈ ℕ |
Colors of variables: wff set class |
Syntax hints: ∈ wcel 1465 (class class class)co 5742 1c1 7589 + caddc 7591 ℕcn 8688 2c2 8739 |
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-sep 4016 ax-cnex 7679 ax-resscn 7680 ax-1re 7682 ax-addrcl 7685 |
This theorem depends on definitions: df-bi 116 df-3an 949 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-rex 2399 df-v 2662 df-un 3045 df-in 3047 df-ss 3054 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-int 3742 df-br 3900 df-iota 5058 df-fv 5101 df-ov 5745 df-inn 8689 df-2 8747 |
This theorem is referenced by: 3nn 8850 2nn0 8962 2z 9050 uz3m2nn 9336 ige2m1fz1 9857 qbtwnre 10002 flhalf 10043 sqeq0 10324 sqeq0d 10391 facavg 10460 bcn2 10478 resqrexlemnm 10758 abs00ap 10802 geo2sum 11251 geo2lim 11253 ege2le3 11304 ef01bndlem 11390 mod2eq0even 11502 mod2eq1n2dvds 11503 sqgcd 11644 3lcm2e6woprm 11694 prm2orodd 11734 3prm 11736 4nprm 11737 divgcdodd 11748 isevengcd2 11763 3lcm2e6 11765 pw2dvdslemn 11770 pw2dvds 11771 pw2dvdseulemle 11772 oddpwdclemxy 11774 oddpwdclemodd 11777 oddpwdclemdc 11778 oddpwdc 11779 sqpweven 11780 2sqpwodd 11781 evenennn 11833 exmidunben 11866 plusgndx 11979 plusgid 11980 plusgslid 11981 grpstrg 11993 grpbaseg 11994 grpplusgg 11995 rngstrg 12001 lmodstrd 12019 topgrpstrd 12037 dsndx 12044 dsid 12045 dsslid 12046 dveflem 12782 ex-fl 12864 ex-ceil 12865 |
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