Theorem cardsucinf

Description: The cardinality of the successor of an infinite ordinal. (Contributed by Mario Carneiro, 11-Jan-2013)

	Ref	Expression
Assertion	cardsucinf	⊢ ( ( 𝐴 ∈ On ∧ ω ⊆ 𝐴 ) → ( card ‘ suc 𝐴 ) = ( card ‘ 𝐴 ) )

Proof

Step	Hyp	Ref	Expression
1		infensuc	⊢ ( ( 𝐴 ∈ On ∧ ω ⊆ 𝐴 ) → 𝐴 ≈ suc 𝐴 )
2		carden2b	⊢ ( 𝐴 ≈ suc 𝐴 → ( card ‘ 𝐴 ) = ( card ‘ suc 𝐴 ) )
3	1 2	syl	⊢ ( ( 𝐴 ∈ On ∧ ω ⊆ 𝐴 ) → ( card ‘ 𝐴 ) = ( card ‘ suc 𝐴 ) )
4	3	eqcomd	⊢ ( ( 𝐴 ∈ On ∧ ω ⊆ 𝐴 ) → ( card ‘ suc 𝐴 ) = ( card ‘ 𝐴 ) )