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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 7916 | . . . . . 6 | |
2 | 1 | eleq2i 2104 | . . . . 5 |
3 | elintg 3623 | . . . . 5 | |
4 | 2, 3 | syl5bb 181 | . . . 4 |
5 | 4 | ibi 165 | . . 3 |
6 | vex 2560 | . . . . . . . 8 | |
7 | eleq2 2101 | . . . . . . . . 9 | |
8 | eleq2 2101 | . . . . . . . . . 10 | |
9 | 8 | raleqbi1dv 2513 | . . . . . . . . 9 |
10 | 7, 9 | anbi12d 442 | . . . . . . . 8 |
11 | 6, 10 | elab 2687 | . . . . . . 7 |
12 | 11 | simprbi 260 | . . . . . 6 |
13 | oveq1 5519 | . . . . . . . 8 | |
14 | 13 | eleq1d 2106 | . . . . . . 7 |
15 | 14 | rspcva 2654 | . . . . . 6 |
16 | 12, 15 | sylan2 270 | . . . . 5 |
17 | 16 | expcom 109 | . . . 4 |
18 | 17 | ralimia 2382 | . . 3 |
19 | 5, 18 | syl 14 | . 2 |
20 | nnre 7921 | . . . 4 | |
21 | 1red 7042 | . . . 4 | |
22 | 20, 21 | readdcld 7055 | . . 3 |
23 | 1 | eleq2i 2104 | . . . 4 |
24 | elintg 3623 | . . . 4 | |
25 | 23, 24 | syl5bb 181 | . . 3 |
26 | 22, 25 | syl 14 | . 2 |
27 | 19, 26 | mpbird 156 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 97 wb 98 wceq 1243 wcel 1393 cab 2026 wral 2306 cint 3615 (class class class)co 5512 cr 6888 c1 6890 caddc 6892 cn 7914 |
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-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-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 ax-cnex 6975 ax-resscn 6976 ax-1re 6978 ax-addrcl 6981 |
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-ral 2311 df-rex 2312 df-v 2559 df-un 2922 df-in 2924 df-ss 2931 df-sn 3381 df-pr 3382 df-op 3384 df-uni 3581 df-int 3616 df-br 3765 df-iota 4867 df-fv 4910 df-ov 5515 df-inn 7915 |
This theorem is referenced by: peano2nnd 7929 nnind 7930 nnaddcl 7934 2nn 8077 3nn 8078 4nn 8079 5nn 8080 6nn 8081 7nn 8082 8nn 8083 9nn 8084 10nn 8085 nneoor 8340 fzonn0p1p1 9069 expp1 9262 resqrexlemfp1 9607 resqrexlemcalc3 9614 sqrt2irr 9878 |
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