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Mirrors > Home > ILE Home > Th. List > peano2nn | Unicode version |
Description: Peano postulate: a successor of a positive integer is a positive integer. (Contributed by NM, 11-Jan-1997.) (Revised by Mario Carneiro, 17-Nov-2014.) |
Ref | Expression |
---|---|
peano2nn |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfnn2 8722 | . . . . . 6 | |
2 | 1 | eleq2i 2206 | . . . . 5 |
3 | elintg 3779 | . . . . 5 | |
4 | 2, 3 | syl5bb 191 | . . . 4 |
5 | 4 | ibi 175 | . . 3 |
6 | vex 2689 | . . . . . . . 8 | |
7 | eleq2 2203 | . . . . . . . . 9 | |
8 | eleq2 2203 | . . . . . . . . . 10 | |
9 | 8 | raleqbi1dv 2634 | . . . . . . . . 9 |
10 | 7, 9 | anbi12d 464 | . . . . . . . 8 |
11 | 6, 10 | elab 2828 | . . . . . . 7 |
12 | 11 | simprbi 273 | . . . . . 6 |
13 | oveq1 5781 | . . . . . . . 8 | |
14 | 13 | eleq1d 2208 | . . . . . . 7 |
15 | 14 | rspcva 2787 | . . . . . 6 |
16 | 12, 15 | sylan2 284 | . . . . 5 |
17 | 16 | expcom 115 | . . . 4 |
18 | 17 | ralimia 2493 | . . 3 |
19 | 5, 18 | syl 14 | . 2 |
20 | nnre 8727 | . . . 4 | |
21 | 1red 7781 | . . . 4 | |
22 | 20, 21 | readdcld 7795 | . . 3 |
23 | 1 | eleq2i 2206 | . . . 4 |
24 | elintg 3779 | . . . 4 | |
25 | 23, 24 | syl5bb 191 | . . 3 |
26 | 22, 25 | syl 14 | . 2 |
27 | 19, 26 | mpbird 166 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wceq 1331 wcel 1480 cab 2125 wral 2416 cint 3771 (class class class)co 5774 cr 7619 c1 7621 caddc 7623 cn 8720 |
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 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-cnex 7711 ax-resscn 7712 ax-1re 7714 ax-addrcl 7717 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-rex 2422 df-v 2688 df-un 3075 df-in 3077 df-ss 3084 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-int 3772 df-br 3930 df-iota 5088 df-fv 5131 df-ov 5777 df-inn 8721 |
This theorem is referenced by: peano2nnd 8735 nnind 8736 nnaddcl 8740 2nn 8881 3nn 8882 4nn 8883 5nn 8884 6nn 8885 7nn 8886 8nn 8887 9nn 8888 nneoor 9153 10nn 9197 nnsplit 9914 fzonn0p1p1 9990 expp1 10300 facp1 10476 resqrexlemfp1 10781 resqrexlemcalc3 10788 trireciplem 11269 trirecip 11270 cvgratnnlemnexp 11293 cvgratz 11301 nno 11603 nnoddm1d2 11607 rplpwr 11715 prmind2 11801 sqrt2irr 11840 |
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