Theorem ldgenpisyslem3

Description: Lemma for ldgenpisys . (Contributed by Thierry Arnoux, 18-Jul-2020)

	Ref	Expression
Hypotheses	dynkin.p	$$⊢P=\left\{s\in 𝒫𝒫O|\mathrm{fi}\left(s\right)\subseteq s\right\}$$
	dynkin.l	$$⊢L=\left\{s\in 𝒫𝒫O|\left(\varnothing \in s\wedge \forall x\in sO\setminus x\in s\wedge \forall x\in 𝒫s\left(\left(x\preccurlyeq \mathrm{\omega }\wedge \underset{y\in x}{Disj}y\right)\to \bigcup x\in s\right)\right)\right\}$$
	dynkin.o	$$⊢\phi \to O\in V$$
	ldgenpisys.e	$$⊢E=\bigcap \left\{t\in L|T\subseteq t\right\}$$
	ldgenpisys.1	$$⊢\phi \to T\in P$$
	ldgenpisyslem3.1	$$⊢\phi \to A\in T$$
Assertion	ldgenpisyslem3	$$⊢\phi \to E\subseteq \left\{b\in 𝒫O|A\cap b\in E\right\}$$

Proof

Step	Hyp	Ref	Expression
1		dynkin.p	$$⊢P=\left\{s\in 𝒫𝒫O|\mathrm{fi}\left(s\right)\subseteq s\right\}$$
2		dynkin.l	$$⊢L=\left\{s\in 𝒫𝒫O|\left(\varnothing \in s\wedge \forall x\in sO\setminus x\in s\wedge \forall x\in 𝒫s\left(\left(x\preccurlyeq \mathrm{\omega }\wedge \underset{y\in x}{Disj}y\right)\to \bigcup x\in s\right)\right)\right\}$$
3		dynkin.o	$$⊢\phi \to O\in V$$
4		ldgenpisys.e	$$⊢E=\bigcap \left\{t\in L|T\subseteq t\right\}$$
5		ldgenpisys.1	$$⊢\phi \to T\in P$$
6		ldgenpisyslem3.1	$$⊢\phi \to A\in T$$
7		id	$$⊢T\subseteq t\to T\subseteq t$$
8	7	rgenw	$$⊢\forall t\in L\left(T\subseteq t\to T\subseteq t\right)$$
9		ssintrab	$$⊢T\subseteq \bigcap \left\{t\in L|T\subseteq t\right\}\leftrightarrow \forall t\in L\left(T\subseteq t\to T\subseteq t\right)$$
10	8 9	mpbir	$$⊢T\subseteq \bigcap \left\{t\in L|T\subseteq t\right\}$$
11	10 4	sseqtrri	$$⊢T\subseteq E$$
12	11 6	sselid	$$⊢\phi \to A\in E$$
13	1	ispisys	$$⊢T\in P\leftrightarrow \left(T\in 𝒫𝒫O\wedge \mathrm{fi}\left(T\right)\subseteq T\right)$$
14	5 13	sylib	$$⊢\phi \to \left(T\in 𝒫𝒫O\wedge \mathrm{fi}\left(T\right)\subseteq T\right)$$
15	14	simpld	$$⊢\phi \to T\in 𝒫𝒫O$$
16		elpwi	$$⊢T\in 𝒫𝒫O\to T\subseteq 𝒫O$$
17	15 16	syl	$$⊢\phi \to T\subseteq 𝒫O$$
18	5	adantr	$$⊢\left(\phi \wedge b\in T\right)\to T\in P$$
19	6	adantr	$$⊢\left(\phi \wedge b\in T\right)\to A\in T$$
20		simpr	$$⊢\left(\phi \wedge b\in T\right)\to b\in T$$
21	1	inelpisys	$$⊢\left(T\in P\wedge A\in T\wedge b\in T\right)\to A\cap b\in T$$
22	18 19 20 21	syl3anc	$$⊢\left(\phi \wedge b\in T\right)\to A\cap b\in T$$
23	11 22	sselid	$$⊢\left(\phi \wedge b\in T\right)\to A\cap b\in E$$
24	23	ralrimiva	$$⊢\phi \to \forall b\in TA\cap b\in E$$
25	17 24	jca	$$⊢\phi \to \left(T\subseteq 𝒫O\wedge \forall b\in TA\cap b\in E\right)$$
26		ssrab	$$⊢T\subseteq \left\{b\in 𝒫O|A\cap b\in E\right\}\leftrightarrow \left(T\subseteq 𝒫O\wedge \forall b\in TA\cap b\in E\right)$$
27	25 26	sylibr	$$⊢\phi \to T\subseteq \left\{b\in 𝒫O|A\cap b\in E\right\}$$
28	1 2 3 4 5 12 27	ldgenpisyslem2	$$⊢\phi \to E\subseteq \left\{b\in 𝒫O|A\cap b\in E\right\}$$