cdleme3g

Description: Part of proof of Lemma E in Crawley p. 113. Lemma leading to cdleme3fa and cdleme3 . (Contributed by NM, 7-Jun-2012)

	Ref	Expression
Hypotheses	cdleme1.l	$⊢ \underset{˙}{\leq} = \leq_{K}$
	cdleme1.j	$⊢ \underset{˙}{\lor} = join (K)$
	cdleme1.m	$⊢ \underset{˙}{\land} = meet (K)$
	cdleme1.a	$⊢ A = Atoms (K)$
	cdleme1.h	$⊢ H = LHyp (K)$
	cdleme1.u	$⊢ U = (P \underset{˙}{\lor} Q) \underset{˙}{\land} W$
	cdleme1.f	$⊢ F = (R \underset{˙}{\lor} U) \underset{˙}{\land} (Q \underset{˙}{\lor} ((P \underset{˙}{\lor} R) \underset{˙}{\land} W))$
	cdleme3.3	$⊢ V = (P \underset{˙}{\lor} R) \underset{˙}{\land} W$
Assertion	cdleme3g	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land (P \neq Q \land \neg R \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to F \neq U$

Proof

Step	Hyp	Ref	Expression
1		cdleme1.l	$⊢ \underset{˙}{\leq} = \leq_{K}$
2		cdleme1.j	$⊢ \underset{˙}{\lor} = join (K)$
3		cdleme1.m	$⊢ \underset{˙}{\land} = meet (K)$
4		cdleme1.a	$⊢ A = Atoms (K)$
5		cdleme1.h	$⊢ H = LHyp (K)$
6		cdleme1.u	$⊢ U = (P \underset{˙}{\lor} Q) \underset{˙}{\land} W$
7		cdleme1.f	$⊢ F = (R \underset{˙}{\lor} U) \underset{˙}{\land} (Q \underset{˙}{\lor} ((P \underset{˙}{\lor} R) \underset{˙}{\land} W))$
8		cdleme3.3	$⊢ V = (P \underset{˙}{\lor} R) \underset{˙}{\land} W$
9	1 2 3 4 5 6 7 8	cdleme3d	$⊢ F = (R \underset{˙}{\lor} U) \underset{˙}{\land} (Q \underset{˙}{\lor} V)$
10		simp1l	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land (P \neq Q \land \neg R \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to K \in HL$
11	10	hllatd	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land (P \neq Q \land \neg R \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to K \in Lat$
12		simp23l	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land (P \neq Q \land \neg R \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to R \in A$
13		simp1	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land (P \neq Q \land \neg R \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to (K \in HL \land W \in H)$
14		simp21	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land (P \neq Q \land \neg R \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to (P \in A \land \neg P \underset{˙}{\leq} W)$
15		simp22l	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land (P \neq Q \land \neg R \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to Q \in A$
16		simp3l	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land (P \neq Q \land \neg R \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to P \neq Q$
17	1 2 3 4 5 6	lhpat2	$⊢ ((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land P \neq Q)) \to U \in A$
18	13 14 15 16 17	syl112anc	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land (P \neq Q \land \neg R \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to U \in A$
19		eqid	$⊢ {Base}_{K} = {Base}_{K}$
20	19 2 4	hlatjcl	$⊢ (K \in HL \land R \in A \land U \in A) \to R \underset{˙}{\lor} U \in {Base}_{K}$
21	10 12 18 20	syl3anc	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land (P \neq Q \land \neg R \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to R \underset{˙}{\lor} U \in {Base}_{K}$
22		simp3r	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land (P \neq Q \land \neg R \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to \neg R \underset{˙}{\leq} (P \underset{˙}{\lor} Q)$
23	12 22	jca	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land (P \neq Q \land \neg R \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to (R \in A \land \neg R \underset{˙}{\leq} (P \underset{˙}{\lor} Q))$
24	1 2 3 4 5 6 7 8	cdleme3e	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land Q \in A \land (R \in A \land \neg R \underset{˙}{\leq} (P \underset{˙}{\lor} Q)))) \to V \in A$
25	13 14 15 23 24	syl13anc	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land (P \neq Q \land \neg R \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to V \in A$
26	19 2 4	hlatjcl	$⊢ (K \in HL \land Q \in A \land V \in A) \to Q \underset{˙}{\lor} V \in {Base}_{K}$
27	10 15 25 26	syl3anc	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land (P \neq Q \land \neg R \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to Q \underset{˙}{\lor} V \in {Base}_{K}$
28	19 1 3	latmle2	$⊢ (K \in Lat \land R \underset{˙}{\lor} U \in {Base}_{K} \land Q \underset{˙}{\lor} V \in {Base}_{K}) \to ((R \underset{˙}{\lor} U) \underset{˙}{\land} (Q \underset{˙}{\lor} V)) \underset{˙}{\leq} (Q \underset{˙}{\lor} V)$
29	11 21 27 28	syl3anc	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land (P \neq Q \land \neg R \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to ((R \underset{˙}{\lor} U) \underset{˙}{\land} (Q \underset{˙}{\lor} V)) \underset{˙}{\leq} (Q \underset{˙}{\lor} V)$
30	9 29	eqbrtrid	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land (P \neq Q \land \neg R \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to F \underset{˙}{\leq} (Q \underset{˙}{\lor} V)$
31		simp22r	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land (P \neq Q \land \neg R \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to \neg Q \underset{˙}{\leq} W$
32		simp23	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land (P \neq Q \land \neg R \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to (R \in A \land \neg R \underset{˙}{\leq} W)$
33		simp3	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land (P \neq Q \land \neg R \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to (P \neq Q \land \neg R \underset{˙}{\leq} (P \underset{˙}{\lor} Q))$
34	1 2 3 4 5 6 8	cdleme0e	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land Q \in A \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land (P \neq Q \land \neg R \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to U \neq V$
35	13 14 15 32 33 34	syl131anc	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land (P \neq Q \land \neg R \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to U \neq V$
36	1 2 4	hlatexch2	$⊢ (K \in HL \land (U \in A \land Q \in A \land V \in A) \land U \neq V) \to (U \underset{˙}{\leq} (Q \underset{˙}{\lor} V) \to Q \underset{˙}{\leq} (U \underset{˙}{\lor} V))$
37	10 18 15 25 35 36	syl131anc	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land (P \neq Q \land \neg R \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to (U \underset{˙}{\leq} (Q \underset{˙}{\lor} V) \to Q \underset{˙}{\leq} (U \underset{˙}{\lor} V))$
38		simp21l	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land (P \neq Q \land \neg R \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to P \in A$
39	19 2 4	hlatjcl	$⊢ (K \in HL \land P \in A \land Q \in A) \to P \underset{˙}{\lor} Q \in {Base}_{K}$
40	10 38 15 39	syl3anc	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land (P \neq Q \land \neg R \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to P \underset{˙}{\lor} Q \in {Base}_{K}$
41		simp1r	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land (P \neq Q \land \neg R \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to W \in H$
42	19 5	lhpbase	$⊢ W \in H \to W \in {Base}_{K}$
43	41 42	syl	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land (P \neq Q \land \neg R \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to W \in {Base}_{K}$
44	19 1 3	latmle2	$⊢ (K \in Lat \land P \underset{˙}{\lor} Q \in {Base}_{K} \land W \in {Base}_{K}) \to ((P \underset{˙}{\lor} Q) \underset{˙}{\land} W) \underset{˙}{\leq} W$
45	11 40 43 44	syl3anc	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land (P \neq Q \land \neg R \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to ((P \underset{˙}{\lor} Q) \underset{˙}{\land} W) \underset{˙}{\leq} W$
46	6 45	eqbrtrid	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land (P \neq Q \land \neg R \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to U \underset{˙}{\leq} W$
47	19 2 4	hlatjcl	$⊢ (K \in HL \land P \in A \land R \in A) \to P \underset{˙}{\lor} R \in {Base}_{K}$
48	10 38 12 47	syl3anc	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land (P \neq Q \land \neg R \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to P \underset{˙}{\lor} R \in {Base}_{K}$
49	19 1 3	latmle2	$⊢ (K \in Lat \land P \underset{˙}{\lor} R \in {Base}_{K} \land W \in {Base}_{K}) \to ((P \underset{˙}{\lor} R) \underset{˙}{\land} W) \underset{˙}{\leq} W$
50	11 48 43 49	syl3anc	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land (P \neq Q \land \neg R \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to ((P \underset{˙}{\lor} R) \underset{˙}{\land} W) \underset{˙}{\leq} W$
51	8 50	eqbrtrid	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land (P \neq Q \land \neg R \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to V \underset{˙}{\leq} W$
52	19 4	atbase	$⊢ U \in A \to U \in {Base}_{K}$
53	18 52	syl	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land (P \neq Q \land \neg R \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to U \in {Base}_{K}$
54	19 4	atbase	$⊢ V \in A \to V \in {Base}_{K}$
55	25 54	syl	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land (P \neq Q \land \neg R \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to V \in {Base}_{K}$
56	19 1 2	latjle12	$⊢ (K \in Lat \land (U \in {Base}_{K} \land V \in {Base}_{K} \land W \in {Base}_{K})) \to ((U \underset{˙}{\leq} W \land V \underset{˙}{\leq} W) \leftrightarrow (U \underset{˙}{\lor} V) \underset{˙}{\leq} W)$
57	11 53 55 43 56	syl13anc	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land (P \neq Q \land \neg R \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to ((U \underset{˙}{\leq} W \land V \underset{˙}{\leq} W) \leftrightarrow (U \underset{˙}{\lor} V) \underset{˙}{\leq} W)$
58	46 51 57	mpbi2and	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land (P \neq Q \land \neg R \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to (U \underset{˙}{\lor} V) \underset{˙}{\leq} W$
59	19 4	atbase	$⊢ Q \in A \to Q \in {Base}_{K}$
60	15 59	syl	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land (P \neq Q \land \neg R \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to Q \in {Base}_{K}$
61	19 2 4	hlatjcl	$⊢ (K \in HL \land U \in A \land V \in A) \to U \underset{˙}{\lor} V \in {Base}_{K}$
62	10 18 25 61	syl3anc	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land (P \neq Q \land \neg R \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to U \underset{˙}{\lor} V \in {Base}_{K}$
63	19 1	lattr	$⊢ (K \in Lat \land (Q \in {Base}_{K} \land U \underset{˙}{\lor} V \in {Base}_{K} \land W \in {Base}_{K})) \to ((Q \underset{˙}{\leq} (U \underset{˙}{\lor} V) \land (U \underset{˙}{\lor} V) \underset{˙}{\leq} W) \to Q \underset{˙}{\leq} W)$
64	11 60 62 43 63	syl13anc	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land (P \neq Q \land \neg R \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to ((Q \underset{˙}{\leq} (U \underset{˙}{\lor} V) \land (U \underset{˙}{\lor} V) \underset{˙}{\leq} W) \to Q \underset{˙}{\leq} W)$
65	58 64	mpan2d	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land (P \neq Q \land \neg R \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to (Q \underset{˙}{\leq} (U \underset{˙}{\lor} V) \to Q \underset{˙}{\leq} W)$
66	37 65	syld	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land (P \neq Q \land \neg R \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to (U \underset{˙}{\leq} (Q \underset{˙}{\lor} V) \to Q \underset{˙}{\leq} W)$
67	31 66	mtod	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land (P \neq Q \land \neg R \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to \neg U \underset{˙}{\leq} (Q \underset{˙}{\lor} V)$
68		nbrne2	$⊢ (F \underset{˙}{\leq} (Q \underset{˙}{\lor} V) \land \neg U \underset{˙}{\leq} (Q \underset{˙}{\lor} V)) \to F \neq U$
69	30 67 68	syl2anc	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land (P \neq Q \land \neg R \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to F \neq U$

Metamath Proof Explorer

Theorem cdleme3g

Proof