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| Mirrors > Home > MPE Home > Th. List > peano3 | Structured version Visualization version 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 | nsuceq0 6467 | . 2 ⊢ suc 𝐴 ≠ ∅ | |
| 2 | 1 | a1i 11 | 1 ⊢ (𝐴 ∈ ω → suc 𝐴 ≠ ∅) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∈ wcel 2108 ≠ wne 2940 ∅c0 4333 suc csuc 6386 ωcom 7887 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-nul 5306 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-v 3482 df-dif 3954 df-un 3956 df-nul 4334 df-sn 4627 df-suc 6390 |
| This theorem is referenced by: (None) |
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