4atexlemwb

	Ref	Expression
Hypotheses	4thatlem.ph	$⊢ φ \leftrightarrow (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (S \in A \land (R \in A \land \neg R \underset{˙}{\leq} W \land P \underset{˙}{\lor} R = Q \underset{˙}{\lor} R) \land (T \in A \land U \underset{˙}{\lor} T = V \underset{˙}{\lor} T)) \land (P \neq Q \land \neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q)))$
	4thatlemmwb.h	$⊢ H = LHyp (K)$
Assertion	4atexlemwb	$⊢ φ \to W \in {Base}_{K}$

Step	Hyp	Ref	Expression
1		4thatlem.ph	$⊢ φ \leftrightarrow (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (S \in A \land (R \in A \land \neg R \underset{˙}{\leq} W \land P \underset{˙}{\lor} R = Q \underset{˙}{\lor} R) \land (T \in A \land U \underset{˙}{\lor} T = V \underset{˙}{\lor} T)) \land (P \neq Q \land \neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q)))$
2		4thatlemmwb.h	$⊢ H = LHyp (K)$
3	1	4atexlemw	$⊢ φ \to W \in H$
4		eqid	$⊢ {Base}_{K} = {Base}_{K}$
5	4 2	lhpbase	$⊢ W \in H \to W \in {Base}_{K}$
6	3 5	syl	$⊢ φ \to W \in {Base}_{K}$

Metamath Proof Explorer