Theorem peano4

Description: Two natural numbers are equal iff their successors are equal, i.e. the successor function is one-to-one. One of Peano's five postulates for arithmetic. Proposition 7.30(4) of TakeutiZaring p. 43. (Contributed by NM, 3-Sep-2003)

	Ref	Expression
Assertion	peano4	$$⊢\left(A\in \mathrm{\omega }\wedge B\in \mathrm{\omega }\right)\to \left(\mathrm{suc}A=\mathrm{suc}B\leftrightarrow A=B\right)$$

Proof

Step	Hyp	Ref	Expression
1		nnon	$$⊢A\in \mathrm{\omega }\to A\in \mathrm{On}$$
2		nnon	$$⊢B\in \mathrm{\omega }\to B\in \mathrm{On}$$
3		suc11	$$⊢\left(A\in \mathrm{On}\wedge B\in \mathrm{On}\right)\to \left(\mathrm{suc}A=\mathrm{suc}B\leftrightarrow A=B\right)$$
4	1 2 3	syl2an	$$⊢\left(A\in \mathrm{\omega }\wedge B\in \mathrm{\omega }\right)\to \left(\mathrm{suc}A=\mathrm{suc}B\leftrightarrow A=B\right)$$