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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 6252 | . 2 ⊢ suc 𝐴 ≠ ∅ | |
2 | 1 | a1i 11 | 1 ⊢ (𝐴 ∈ ω → suc 𝐴 ≠ ∅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2114 ≠ wne 2934 ∅c0 4211 suc csuc 6174 ωcom 7601 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-ext 2710 ax-nul 5174 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-tru 1545 df-fal 1555 df-ex 1787 df-sb 2075 df-clab 2717 df-cleq 2730 df-clel 2811 df-ne 2935 df-v 3400 df-dif 3846 df-un 3848 df-nul 4212 df-sn 4517 df-suc 6178 |
This theorem is referenced by: (None) |
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