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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 4300 | 1 ⊢ (𝐴 ∈ ω → suc 𝐴 ≠ ∅) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 1463 ≠ wne 2282 ∅c0 3329 suc csuc 4247 ωcom 4464 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 586 ax-in2 587 ax-io 681 ax-5 1406 ax-7 1407 ax-gen 1408 ax-ie1 1452 ax-ie2 1453 ax-8 1465 ax-10 1466 ax-11 1467 ax-i12 1468 ax-bndl 1469 ax-4 1470 ax-17 1489 ax-i9 1493 ax-ial 1497 ax-i5r 1498 ax-ext 2097 |
This theorem depends on definitions: df-bi 116 df-tru 1317 df-nf 1420 df-sb 1719 df-clab 2102 df-cleq 2108 df-clel 2111 df-nfc 2244 df-ne 2283 df-v 2659 df-dif 3039 df-un 3041 df-nul 3330 df-sn 3499 df-suc 4253 |
This theorem is referenced by: nndceq0 4491 frecabcl 6250 frecsuclem 6257 nnsucsssuc 6342 php5 6705 findcard2 6736 findcard2s 6737 omp1eomlem 6931 ctmlemr 6945 nnsf 12891 peano4nninf 12892 |
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