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Mirrors > Home > ILE Home > Th. List > peano3 | Unicode version |
Description: The successor of any natural number is not zero. One of Peano's five postulates for arithmetic. Proposition 7.30(3) of [TakeutiZaring] p. 42. (Contributed by NM, 3-Sep-2003.) |
Ref | Expression |
---|---|
peano3 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nsuceq0g 4380 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wcel 2128 wne 2327 c0 3395 csuc 4327 com 4551 |
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-in1 604 ax-in2 605 ax-io 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2139 |
This theorem depends on definitions: df-bi 116 df-tru 1338 df-nf 1441 df-sb 1743 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ne 2328 df-v 2714 df-dif 3104 df-un 3106 df-nul 3396 df-sn 3567 df-suc 4333 |
This theorem is referenced by: nndceq0 4579 frecabcl 6348 frecsuclem 6355 nnsucsssuc 6441 php5 6805 findcard2 6836 findcard2s 6837 omp1eomlem 7040 ctmlemr 7054 nnsf 13648 peano4nninf 13649 |
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