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 ∈ ω → ( card ‘ suc 𝐴 ) = suc ( card ‘ 𝐴 ) )

Proof

Step	Hyp	Ref	Expression
1		peano2	⊢ ( 𝐴 ∈ ω → suc 𝐴 ∈ ω )
2		cardnn	⊢ ( suc 𝐴 ∈ ω → ( card ‘ suc 𝐴 ) = suc 𝐴 )
3	1 2	syl	⊢ ( 𝐴 ∈ ω → ( card ‘ suc 𝐴 ) = suc 𝐴 )
4		cardnn	⊢ ( 𝐴 ∈ ω → ( card ‘ 𝐴 ) = 𝐴 )
5		suceq	⊢ ( ( card ‘ 𝐴 ) = 𝐴 → suc ( card ‘ 𝐴 ) = suc 𝐴 )
6	4 5	syl	⊢ ( 𝐴 ∈ ω → suc ( card ‘ 𝐴 ) = suc 𝐴 )
7	3 6	eqtr4d	⊢ ( 𝐴 ∈ ω → ( card ‘ suc 𝐴 ) = suc ( card ‘ 𝐴 ) )