Theorem winalim

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

	Ref	Expression
Assertion	winalim	$$⊢A\in {\mathrm{Inacc}}_{𝑤}\to \mathrm{Lim}A$$

Proof

Step	Hyp	Ref	Expression
1		winainf	$$⊢A\in {\mathrm{Inacc}}_{𝑤}\to \mathrm{\omega }\subseteq A$$
2		winacard	$$⊢A\in {\mathrm{Inacc}}_{𝑤}\to \mathrm{card}\left(A\right)=A$$
3		cardlim	$$⊢\mathrm{\omega }\subseteq \mathrm{card}\left(A\right)\leftrightarrow \mathrm{Lim}\mathrm{card}\left(A\right)$$
4		sseq2	$$⊢\mathrm{card}\left(A\right)=A\to \left(\mathrm{\omega }\subseteq \mathrm{card}\left(A\right)\leftrightarrow \mathrm{\omega }\subseteq A\right)$$
5		limeq	$$⊢\mathrm{card}\left(A\right)=A\to \left(\mathrm{Lim}\mathrm{card}\left(A\right)\leftrightarrow \mathrm{Lim}A\right)$$
6	4 5	bibi12d	$$⊢\mathrm{card}\left(A\right)=A\to \left(\left(\mathrm{\omega }\subseteq \mathrm{card}\left(A\right)\leftrightarrow \mathrm{Lim}\mathrm{card}\left(A\right)\right)\leftrightarrow \left(\mathrm{\omega }\subseteq A\leftrightarrow \mathrm{Lim}A\right)\right)$$
7	3 6	mpbii	$$⊢\mathrm{card}\left(A\right)=A\to \left(\mathrm{\omega }\subseteq A\leftrightarrow \mathrm{Lim}A\right)$$
8	2 7	syl	$$⊢A\in {\mathrm{Inacc}}_{𝑤}\to \left(\mathrm{\omega }\subseteq A\leftrightarrow \mathrm{Lim}A\right)$$
9	1 8	mpbid	$$⊢A\in {\mathrm{Inacc}}_{𝑤}\to \mathrm{Lim}A$$