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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 5805 | . 2 ⊢ suc 𝐴 ≠ ∅ | |
2 | 1 | a1i 11 | 1 ⊢ (𝐴 ∈ ω → suc 𝐴 ≠ ∅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 1990 ≠ wne 2794 ∅c0 3915 suc csuc 5725 ωcom 7065 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-nul 4789 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-v 3202 df-dif 3577 df-un 3579 df-nul 3916 df-sn 4178 df-suc 5729 |
This theorem is referenced by: (None) |
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