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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 5722 | . 2 ⊢ suc 𝐴 ≠ ∅ | |
2 | 1 | a1i 11 | 1 ⊢ (𝐴 ∈ ω → suc 𝐴 ≠ ∅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 1977 ≠ wne 2780 ∅c0 3874 suc csuc 5642 ωcom 6957 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-nul 4717 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-v 3175 df-dif 3543 df-un 3545 df-nul 3875 df-sn 4126 df-suc 5646 |
This theorem is referenced by: (None) |
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