Theorem php5

Description: Corollary of the Pigeonhole Principle php : a natural number is not equinumerous to its successor. Corollary 10.21(1) of TakeutiZaring p. 90. (Contributed by NM, 26-Jul-2004)

	Ref	Expression
Assertion	php5	$$⊢A\in \mathrm{\omega }\to \neg A\approx \mathrm{suc}A$$

Proof

Step	Hyp	Ref	Expression
1		php4	$$⊢A\in \mathrm{\omega }\to A\prec \mathrm{suc}A$$
2		sdomnen	$$⊢A\prec \mathrm{suc}A\to \neg A\approx \mathrm{suc}A$$
3	1 2	syl	$$⊢A\in \mathrm{\omega }\to \neg A\approx \mathrm{suc}A$$