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Mirrors > Home > ILE Home > Th. List > peano3 | GIF 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 | ⊢ (𝐴 ∈ ω → suc 𝐴 ≠ ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nsuceq0g 4173 | 1 ⊢ (𝐴 ∈ ω → suc 𝐴 ≠ ∅) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 1433 ≠ wne 2245 ∅c0 3251 suc csuc 4120 ωcom 4331 |
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-in1 576 ax-in2 577 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 |
This theorem depends on definitions: df-bi 115 df-tru 1287 df-nf 1390 df-sb 1686 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ne 2246 df-v 2603 df-dif 2975 df-un 2977 df-nul 3252 df-sn 3404 df-suc 4126 |
This theorem is referenced by: nndceq0 4357 frecsuclem3 6013 nnsucsssuc 6094 php5 6344 findcard2 6373 findcard2s 6374 |
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