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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 8041 | . . . . . 6 | |
2 | 1 | eleq2i 2145 | . . . . 5 |
3 | elintg 3644 | . . . . 5 | |
4 | 2, 3 | syl5bb 190 | . . . 4 |
5 | 4 | ibi 174 | . . 3 |
6 | vex 2604 | . . . . . . . 8 | |
7 | eleq2 2142 | . . . . . . . . 9 | |
8 | eleq2 2142 | . . . . . . . . . 10 | |
9 | 8 | raleqbi1dv 2557 | . . . . . . . . 9 |
10 | 7, 9 | anbi12d 456 | . . . . . . . 8 |
11 | 6, 10 | elab 2738 | . . . . . . 7 |
12 | 11 | simprbi 269 | . . . . . 6 |
13 | oveq1 5539 | . . . . . . . 8 | |
14 | 13 | eleq1d 2147 | . . . . . . 7 |
15 | 14 | rspcva 2699 | . . . . . 6 |
16 | 12, 15 | sylan2 280 | . . . . 5 |
17 | 16 | expcom 114 | . . . 4 |
18 | 17 | ralimia 2424 | . . 3 |
19 | 5, 18 | syl 14 | . 2 |
20 | nnre 8046 | . . . 4 | |
21 | 1red 7134 | . . . 4 | |
22 | 20, 21 | readdcld 7148 | . . 3 |
23 | 1 | eleq2i 2145 | . . . 4 |
24 | elintg 3644 | . . . 4 | |
25 | 23, 24 | syl5bb 190 | . . 3 |
26 | 22, 25 | syl 14 | . 2 |
27 | 19, 26 | mpbird 165 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 102 wb 103 wceq 1284 wcel 1433 cab 2067 wral 2348 cint 3636 (class class class)co 5532 cr 6980 c1 6982 caddc 6984 cn 8039 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-cnex 7067 ax-resscn 7068 ax-1re 7070 ax-addrcl 7073 |
This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-nf 1390 df-sb 1686 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ral 2353 df-rex 2354 df-v 2603 df-un 2977 df-in 2979 df-ss 2986 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-int 3637 df-br 3786 df-iota 4887 df-fv 4930 df-ov 5535 df-inn 8040 |
This theorem is referenced by: peano2nnd 8054 nnind 8055 nnaddcl 8059 2nn 8193 3nn 8194 4nn 8195 5nn 8196 6nn 8197 7nn 8198 8nn 8199 9nn 8200 nneoor 8449 10nn 8492 fzonn0p1p1 9222 expp1 9483 facp1 9657 resqrexlemfp1 9895 resqrexlemcalc3 9902 nno 10306 nnoddm1d2 10310 rplpwr 10416 prmind2 10502 sqrt2irr 10541 |
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