Theorem 1ellim

Description: A limit ordinal contains 1. (Contributed by BTernaryTau, 1-Dec-2024)

	Ref	Expression
Assertion	1ellim	$$⊢\mathrm{Lim}A\to 1_{𝑜}\in A$$

Proof

Step	Hyp	Ref	Expression
1		nlim0	$$⊢\neg \mathrm{Lim}\varnothing$$
2		limeq	$$⊢A=\varnothing \to \left(\mathrm{Lim}A\leftrightarrow \mathrm{Lim}\varnothing \right)$$
3	1 2	mtbiri	$$⊢A=\varnothing \to \neg \mathrm{Lim}A$$
4	3	necon2ai	$$⊢\mathrm{Lim}A\to A\ne \varnothing$$
5		nlim1	$$⊢\neg \mathrm{Lim}{1}_{𝑜}$$
6		limeq	$$⊢A=1_{𝑜}\to \left(\mathrm{Lim}A\leftrightarrow \mathrm{Lim}{1}_{𝑜}\right)$$
7	5 6	mtbiri	$$⊢A=1_{𝑜}\to \neg \mathrm{Lim}A$$
8	7	necon2ai	$$⊢\mathrm{Lim}A\to A\ne 1_{𝑜}$$
9		limord	$$⊢\mathrm{Lim}A\to \mathrm{Ord}A$$
10		ord1eln01	$$⊢\mathrm{Ord}A\to \left(1_{𝑜}\in A\leftrightarrow \left(A\ne \varnothing \wedge A\ne 1_{𝑜}\right)\right)$$
11	9 10	syl	$$⊢\mathrm{Lim}A\to \left(1_{𝑜}\in A\leftrightarrow \left(A\ne \varnothing \wedge A\ne 1_{𝑜}\right)\right)$$
12	4 8 11	mpbir2and	$$⊢\mathrm{Lim}A\to 1_{𝑜}\in A$$