Theorem cardsucnn

Description: The cardinality of the successor of a finite ordinal (natural number). This theorem does not hold for infinite ordinals; see cardsucinf . (Contributed by NM, 7-Nov-2008)

	Ref	Expression
Assertion	cardsucnn	$$⊢A\in \mathrm{\omega }\to \mathrm{card}\left(\mathrm{suc}A\right)=\mathrm{suc}\mathrm{card}\left(A\right)$$

Proof

Step	Hyp	Ref	Expression
1		peano2	$$⊢A\in \mathrm{\omega }\to \mathrm{suc}A\in \mathrm{\omega }$$
2		cardnn	$$⊢\mathrm{suc}A\in \mathrm{\omega }\to \mathrm{card}\left(\mathrm{suc}A\right)=\mathrm{suc}A$$
3	1 2	syl	$$⊢A\in \mathrm{\omega }\to \mathrm{card}\left(\mathrm{suc}A\right)=\mathrm{suc}A$$
4		cardnn	$$⊢A\in \mathrm{\omega }\to \mathrm{card}\left(A\right)=A$$
5		suceq	$$⊢\mathrm{card}\left(A\right)=A\to \mathrm{suc}\mathrm{card}\left(A\right)=\mathrm{suc}A$$
6	4 5	syl	$$⊢A\in \mathrm{\omega }\to \mathrm{suc}\mathrm{card}\left(A\right)=\mathrm{suc}A$$
7	3 6	eqtr4d	$$⊢A\in \mathrm{\omega }\to \mathrm{card}\left(\mathrm{suc}A\right)=\mathrm{suc}\mathrm{card}\left(A\right)$$