Theorem winalim

Description: A weakly inaccessible cardinal is a limit ordinal. (Contributed by Mario Carneiro, 29-May-2014)

	Ref	Expression
Assertion	winalim	⊢ ( 𝐴 ∈ Inaccw → Lim 𝐴 )

Proof

Step	Hyp	Ref	Expression
1		winainf	⊢ ( 𝐴 ∈ Inaccw → ω ⊆ 𝐴 )
2		winacard	⊢ ( 𝐴 ∈ Inaccw → ( card ‘ 𝐴 ) = 𝐴 )
3		cardlim	⊢ ( ω ⊆ ( card ‘ 𝐴 ) ↔ Lim ( card ‘ 𝐴 ) )
4		sseq2	⊢ ( ( card ‘ 𝐴 ) = 𝐴 → ( ω ⊆ ( card ‘ 𝐴 ) ↔ ω ⊆ 𝐴 ) )
5		limeq	⊢ ( ( card ‘ 𝐴 ) = 𝐴 → ( Lim ( card ‘ 𝐴 ) ↔ Lim 𝐴 ) )
6	4 5	bibi12d	⊢ ( ( card ‘ 𝐴 ) = 𝐴 → ( ( ω ⊆ ( card ‘ 𝐴 ) ↔ Lim ( card ‘ 𝐴 ) ) ↔ ( ω ⊆ 𝐴 ↔ Lim 𝐴 ) ) )
7	3 6	mpbii	⊢ ( ( card ‘ 𝐴 ) = 𝐴 → ( ω ⊆ 𝐴 ↔ Lim 𝐴 ) )
8	2 7	syl	⊢ ( 𝐴 ∈ Inaccw → ( ω ⊆ 𝐴 ↔ Lim 𝐴 ) )
9	1 8	mpbid	⊢ ( 𝐴 ∈ Inaccw → Lim 𝐴 )