Theorem lhprelat3N 40874

Description: The Hilbert lattice is relatively atomic with respect to co-atoms (lattice hyperplanes). Dual version of hlrelat3 40246. (Contributed by NM, 20-Jun-2012.) (New usage is discouraged.)

Hypotheses

Ref	Expression
lhprelat3.b	⊢ 𝐵 = (Base‘𝐾)
lhprelat3.l	⊢ ≤ = (le‘𝐾)
lhprelat3.s	⊢ < = (lt‘𝐾)
lhprelat3.m	⊢ ∧ = (meet‘𝐾)
lhprelat3.c	⊢ 𝐶 = ( ⋖ ‘𝐾)
lhprelat3.h	⊢ 𝐻 = (LHyp‘𝐾)

Assertion

Ref	Expression
lhprelat3N	⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑋 < 𝑌) → ∃𝑤 ∈ 𝐻 (𝑋 ≤ (𝑌 ∧ 𝑤) ∧ (𝑌 ∧ 𝑤)𝐶𝑌))

Distinct variable groups:   𝑤,𝐶   𝑤,𝐻   𝑤,𝐾   𝑤, ≤   𝑤, ∧   𝑤,𝑋   𝑤,𝑌

Allowed substitution hints:   𝐵(𝑤)   < (𝑤)

Proof of Theorem lhprelat3N

Dummy variable 𝑝 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpr 490	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑋 < 𝑌) ∧ 𝑝 ∈ (Atoms‘𝐾)) → 𝑝 ∈ (Atoms‘𝐾))
2		simpll1 1231	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑋 < 𝑌) ∧ 𝑝 ∈ (Atoms‘𝐾)) → 𝐾 ∈ HL)
3		lhprelat3.b	. . . . . . . 8 ⊢ 𝐵 = (Base‘𝐾)
4		eqid 2765	. . . . . . . 8 ⊢ (Atoms‘𝐾) = (Atoms‘𝐾)
5	3, 4	atbase 40123	. . . . . . 7 ⊢ (𝑝 ∈ (Atoms‘𝐾) → 𝑝 ∈ 𝐵)
6	5	adantl 487	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑋 < 𝑌) ∧ 𝑝 ∈ (Atoms‘𝐾)) → 𝑝 ∈ 𝐵)
7		eqid 2765	. . . . . . 7 ⊢ (oc‘𝐾) = (oc‘𝐾)
8		lhprelat3.h	. . . . . . 7 ⊢ 𝐻 = (LHyp‘𝐾)
9	3, 7, 4, 8	lhpoc2N 40849	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ 𝑝 ∈ 𝐵) → (𝑝 ∈ (Atoms‘𝐾) ↔ ((oc‘𝐾)‘𝑝) ∈ 𝐻))
10	2, 6, 9	syl2anc 596	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑋 < 𝑌) ∧ 𝑝 ∈ (Atoms‘𝐾)) → (𝑝 ∈ (Atoms‘𝐾) ↔ ((oc‘𝐾)‘𝑝) ∈ 𝐻))
11	1, 10	mpbid 235	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑋 < 𝑌) ∧ 𝑝 ∈ (Atoms‘𝐾)) → ((oc‘𝐾)‘𝑝) ∈ 𝐻)
12	11	adantr 486	. . 3 ⊢ (((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑋 < 𝑌) ∧ 𝑝 ∈ (Atoms‘𝐾)) ∧ (((oc‘𝐾)‘𝑌)𝐶(((oc‘𝐾)‘𝑌)(join‘𝐾)𝑝) ∧ (((oc‘𝐾)‘𝑌)(join‘𝐾)𝑝) ≤ ((oc‘𝐾)‘𝑋))) → ((oc‘𝐾)‘𝑝) ∈ 𝐻)
13		hlop 40196	. . . . . . . . 9 ⊢ (𝐾 ∈ HL → 𝐾 ∈ OP)
14	2, 13	syl 18	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑋 < 𝑌) ∧ 𝑝 ∈ (Atoms‘𝐾)) → 𝐾 ∈ OP)
15	2	hllatd 40198	. . . . . . . . 9 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑋 < 𝑌) ∧ 𝑝 ∈ (Atoms‘𝐾)) → 𝐾 ∈ Lat)
16		simpll3 1233	. . . . . . . . 9 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑋 < 𝑌) ∧ 𝑝 ∈ (Atoms‘𝐾)) → 𝑌 ∈ 𝐵)
17	3, 7	opoccl 40028	. . . . . . . . . 10 ⊢ ((𝐾 ∈ OP ∧ 𝑝 ∈ 𝐵) → ((oc‘𝐾)‘𝑝) ∈ 𝐵)
18	14, 6, 17	syl2anc 596	. . . . . . . . 9 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑋 < 𝑌) ∧ 𝑝 ∈ (Atoms‘𝐾)) → ((oc‘𝐾)‘𝑝) ∈ 𝐵)
19		lhprelat3.m	. . . . . . . . . 10 ⊢ ∧ = (meet‘𝐾)
20	3, 19	latmcl 18520	. . . . . . . . 9 ⊢ ((𝐾 ∈ Lat ∧ 𝑌 ∈ 𝐵 ∧ ((oc‘𝐾)‘𝑝) ∈ 𝐵) → (𝑌 ∧ ((oc‘𝐾)‘𝑝)) ∈ 𝐵)
21	15, 16, 18, 20	syl3anc 1398	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑋 < 𝑌) ∧ 𝑝 ∈ (Atoms‘𝐾)) → (𝑌 ∧ ((oc‘𝐾)‘𝑝)) ∈ 𝐵)
22		lhprelat3.c	. . . . . . . . 9 ⊢ 𝐶 = ( ⋖ ‘𝐾)
23	3, 7, 22	cvrcon3b 40111	. . . . . . . 8 ⊢ ((𝐾 ∈ OP ∧ (𝑌 ∧ ((oc‘𝐾)‘𝑝)) ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((𝑌 ∧ ((oc‘𝐾)‘𝑝))𝐶𝑌 ↔ ((oc‘𝐾)‘𝑌)𝐶((oc‘𝐾)‘(𝑌 ∧ ((oc‘𝐾)‘𝑝)))))
24	14, 21, 16, 23	syl3anc 1398	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑋 < 𝑌) ∧ 𝑝 ∈ (Atoms‘𝐾)) → ((𝑌 ∧ ((oc‘𝐾)‘𝑝))𝐶𝑌 ↔ ((oc‘𝐾)‘𝑌)𝐶((oc‘𝐾)‘(𝑌 ∧ ((oc‘𝐾)‘𝑝)))))
25		hlol 40195	. . . . . . . . . 10 ⊢ (𝐾 ∈ HL → 𝐾 ∈ OL)
26	2, 25	syl 18	. . . . . . . . 9 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑋 < 𝑌) ∧ 𝑝 ∈ (Atoms‘𝐾)) → 𝐾 ∈ OL)
27		eqid 2765	. . . . . . . . . 10 ⊢ (join‘𝐾) = (join‘𝐾)
28	3, 27, 19, 7	oldmm3N 40053	. . . . . . . . 9 ⊢ ((𝐾 ∈ OL ∧ 𝑌 ∈ 𝐵 ∧ 𝑝 ∈ 𝐵) → ((oc‘𝐾)‘(𝑌 ∧ ((oc‘𝐾)‘𝑝))) = (((oc‘𝐾)‘𝑌)(join‘𝐾)𝑝))
29	26, 16, 6, 28	syl3anc 1398	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑋 < 𝑌) ∧ 𝑝 ∈ (Atoms‘𝐾)) → ((oc‘𝐾)‘(𝑌 ∧ ((oc‘𝐾)‘𝑝))) = (((oc‘𝐾)‘𝑌)(join‘𝐾)𝑝))
30	29	breq2d 5123	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑋 < 𝑌) ∧ 𝑝 ∈ (Atoms‘𝐾)) → (((oc‘𝐾)‘𝑌)𝐶((oc‘𝐾)‘(𝑌 ∧ ((oc‘𝐾)‘𝑝))) ↔ ((oc‘𝐾)‘𝑌)𝐶(((oc‘𝐾)‘𝑌)(join‘𝐾)𝑝)))
31	24, 30	bitr2d 283	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑋 < 𝑌) ∧ 𝑝 ∈ (Atoms‘𝐾)) → (((oc‘𝐾)‘𝑌)𝐶(((oc‘𝐾)‘𝑌)(join‘𝐾)𝑝) ↔ (𝑌 ∧ ((oc‘𝐾)‘𝑝))𝐶𝑌))
32		simpll2 1232	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑋 < 𝑌) ∧ 𝑝 ∈ (Atoms‘𝐾)) → 𝑋 ∈ 𝐵)
33		lhprelat3.l	. . . . . . . . 9 ⊢ ≤ = (le‘𝐾)
34	3, 33, 7	oplecon3b 40034	. . . . . . . 8 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ∧ (𝑌 ∧ ((oc‘𝐾)‘𝑝)) ∈ 𝐵) → (𝑋 ≤ (𝑌 ∧ ((oc‘𝐾)‘𝑝)) ↔ ((oc‘𝐾)‘(𝑌 ∧ ((oc‘𝐾)‘𝑝))) ≤ ((oc‘𝐾)‘𝑋)))
35	14, 32, 21, 34	syl3anc 1398	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑋 < 𝑌) ∧ 𝑝 ∈ (Atoms‘𝐾)) → (𝑋 ≤ (𝑌 ∧ ((oc‘𝐾)‘𝑝)) ↔ ((oc‘𝐾)‘(𝑌 ∧ ((oc‘𝐾)‘𝑝))) ≤ ((oc‘𝐾)‘𝑋)))
36	29	breq1d 5121	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑋 < 𝑌) ∧ 𝑝 ∈ (Atoms‘𝐾)) → (((oc‘𝐾)‘(𝑌 ∧ ((oc‘𝐾)‘𝑝))) ≤ ((oc‘𝐾)‘𝑋) ↔ (((oc‘𝐾)‘𝑌)(join‘𝐾)𝑝) ≤ ((oc‘𝐾)‘𝑋)))
37	35, 36	bitr2d 283	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑋 < 𝑌) ∧ 𝑝 ∈ (Atoms‘𝐾)) → ((((oc‘𝐾)‘𝑌)(join‘𝐾)𝑝) ≤ ((oc‘𝐾)‘𝑋) ↔ 𝑋 ≤ (𝑌 ∧ ((oc‘𝐾)‘𝑝))))
38	31, 37	anbi12d 644	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑋 < 𝑌) ∧ 𝑝 ∈ (Atoms‘𝐾)) → ((((oc‘𝐾)‘𝑌)𝐶(((oc‘𝐾)‘𝑌)(join‘𝐾)𝑝) ∧ (((oc‘𝐾)‘𝑌)(join‘𝐾)𝑝) ≤ ((oc‘𝐾)‘𝑋)) ↔ ((𝑌 ∧ ((oc‘𝐾)‘𝑝))𝐶𝑌 ∧ 𝑋 ≤ (𝑌 ∧ ((oc‘𝐾)‘𝑝)))))
39	38	biimpa 482	. . . 4 ⊢ (((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑋 < 𝑌) ∧ 𝑝 ∈ (Atoms‘𝐾)) ∧ (((oc‘𝐾)‘𝑌)𝐶(((oc‘𝐾)‘𝑌)(join‘𝐾)𝑝) ∧ (((oc‘𝐾)‘𝑌)(join‘𝐾)𝑝) ≤ ((oc‘𝐾)‘𝑋))) → ((𝑌 ∧ ((oc‘𝐾)‘𝑝))𝐶𝑌 ∧ 𝑋 ≤ (𝑌 ∧ ((oc‘𝐾)‘𝑝))))
40	39	ancomd 467	. . 3 ⊢ (((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑋 < 𝑌) ∧ 𝑝 ∈ (Atoms‘𝐾)) ∧ (((oc‘𝐾)‘𝑌)𝐶(((oc‘𝐾)‘𝑌)(join‘𝐾)𝑝) ∧ (((oc‘𝐾)‘𝑌)(join‘𝐾)𝑝) ≤ ((oc‘𝐾)‘𝑋))) → (𝑋 ≤ (𝑌 ∧ ((oc‘𝐾)‘𝑝)) ∧ (𝑌 ∧ ((oc‘𝐾)‘𝑝))𝐶𝑌))
41		oveq2 7427	. . . . . 6 ⊢ (𝑤 = ((oc‘𝐾)‘𝑝) → (𝑌 ∧ 𝑤) = (𝑌 ∧ ((oc‘𝐾)‘𝑝)))
42	41	breq2d 5123	. . . . 5 ⊢ (𝑤 = ((oc‘𝐾)‘𝑝) → (𝑋 ≤ (𝑌 ∧ 𝑤) ↔ 𝑋 ≤ (𝑌 ∧ ((oc‘𝐾)‘𝑝))))
43	41	breq1d 5121	. . . . 5 ⊢ (𝑤 = ((oc‘𝐾)‘𝑝) → ((𝑌 ∧ 𝑤)𝐶𝑌 ↔ (𝑌 ∧ ((oc‘𝐾)‘𝑝))𝐶𝑌))
44	42, 43	anbi12d 644	. . . 4 ⊢ (𝑤 = ((oc‘𝐾)‘𝑝) → ((𝑋 ≤ (𝑌 ∧ 𝑤) ∧ (𝑌 ∧ 𝑤)𝐶𝑌) ↔ (𝑋 ≤ (𝑌 ∧ ((oc‘𝐾)‘𝑝)) ∧ (𝑌 ∧ ((oc‘𝐾)‘𝑝))𝐶𝑌)))
45	44	rspcev 3583	. . 3 ⊢ ((((oc‘𝐾)‘𝑝) ∈ 𝐻 ∧ (𝑋 ≤ (𝑌 ∧ ((oc‘𝐾)‘𝑝)) ∧ (𝑌 ∧ ((oc‘𝐾)‘𝑝))𝐶𝑌)) → ∃𝑤 ∈ 𝐻 (𝑋 ≤ (𝑌 ∧ 𝑤) ∧ (𝑌 ∧ 𝑤)𝐶𝑌))
46	12, 40, 45	syl2anc 596	. 2 ⊢ (((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑋 < 𝑌) ∧ 𝑝 ∈ (Atoms‘𝐾)) ∧ (((oc‘𝐾)‘𝑌)𝐶(((oc‘𝐾)‘𝑌)(join‘𝐾)𝑝) ∧ (((oc‘𝐾)‘𝑌)(join‘𝐾)𝑝) ≤ ((oc‘𝐾)‘𝑋))) → ∃𝑤 ∈ 𝐻 (𝑋 ≤ (𝑌 ∧ 𝑤) ∧ (𝑌 ∧ 𝑤)𝐶𝑌))
47		simpl1 1210	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑋 < 𝑌) → 𝐾 ∈ HL)
48	47, 13	syl 18	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑋 < 𝑌) → 𝐾 ∈ OP)
49		simpl3 1212	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑋 < 𝑌) → 𝑌 ∈ 𝐵)
50	3, 7	opoccl 40028	. . . 4 ⊢ ((𝐾 ∈ OP ∧ 𝑌 ∈ 𝐵) → ((oc‘𝐾)‘𝑌) ∈ 𝐵)
51	48, 49, 50	syl2anc 596	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑋 < 𝑌) → ((oc‘𝐾)‘𝑌) ∈ 𝐵)
52		simpl2 1211	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑋 < 𝑌) → 𝑋 ∈ 𝐵)
53	3, 7	opoccl 40028	. . . 4 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵) → ((oc‘𝐾)‘𝑋) ∈ 𝐵)
54	48, 52, 53	syl2anc 596	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑋 < 𝑌) → ((oc‘𝐾)‘𝑋) ∈ 𝐵)
55		simpr 490	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑋 < 𝑌) → 𝑋 < 𝑌)
56		lhprelat3.s	. . . . . 6 ⊢ < = (lt‘𝐾)
57	3, 56, 7	opltcon3b 40038	. . . . 5 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 < 𝑌 ↔ ((oc‘𝐾)‘𝑌) < ((oc‘𝐾)‘𝑋)))
58	48, 52, 49, 57	syl3anc 1398	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑋 < 𝑌) → (𝑋 < 𝑌 ↔ ((oc‘𝐾)‘𝑌) < ((oc‘𝐾)‘𝑋)))
59	55, 58	mpbid 235	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑋 < 𝑌) → ((oc‘𝐾)‘𝑌) < ((oc‘𝐾)‘𝑋))
60	3, 33, 56, 27, 22, 4	hlrelat3 40246	. . 3 ⊢ (((𝐾 ∈ HL ∧ ((oc‘𝐾)‘𝑌) ∈ 𝐵 ∧ ((oc‘𝐾)‘𝑋) ∈ 𝐵) ∧ ((oc‘𝐾)‘𝑌) < ((oc‘𝐾)‘𝑋)) → ∃𝑝 ∈ (Atoms‘𝐾)(((oc‘𝐾)‘𝑌)𝐶(((oc‘𝐾)‘𝑌)(join‘𝐾)𝑝) ∧ (((oc‘𝐾)‘𝑌)(join‘𝐾)𝑝) ≤ ((oc‘𝐾)‘𝑋)))
61	47, 51, 54, 59, 60	syl31anc 1400	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑋 < 𝑌) → ∃𝑝 ∈ (Atoms‘𝐾)(((oc‘𝐾)‘𝑌)𝐶(((oc‘𝐾)‘𝑌)(join‘𝐾)𝑝) ∧ (((oc‘𝐾)‘𝑌)(join‘𝐾)𝑝) ≤ ((oc‘𝐾)‘𝑋)))
62	46, 61	r19.29a 3175	1 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑋 < 𝑌) → ∃𝑤 ∈ 𝐻 (𝑋 ≤ (𝑌 ∧ 𝑤) ∧ (𝑌 ∧ 𝑤)𝐶𝑌))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2146  ∃wrex 3091   class class class wbr 5111  ‘cfv 6540  (class class class)co 7419  Basecbs 17293  lecple 17341  occoc 17342  ltcplt 18388  joincjn 18391  meetcmee 18392  Latclat 18511  OPcops 40006  OLcol 40008   ⋖ ccvr 40096  Atomscatm 40097  HLchlt 40184  LHypclh 40818

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-rep 5240  ax-sep 5259  ax-nul 5271  ax-pow 5338  ax-pr 5406  ax-un 7742

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3082  df-rex 3092  df-rmo 3371  df-reu 3372  df-rab 3419  df-v 3459  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-iun 4960  df-br 5112  df-opab 5176  df-mpt 5195  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-riota 7376  df-ov 7422  df-oprab 7423  df-proset 18374  df-poset 18393  df-plt 18408  df-lub 18424  df-glb 18425  df-join 18426  df-meet 18427  df-p0 18503  df-p1 18504  df-lat 18512  df-clat 18579  df-oposet 40010  df-ol 40012  df-oml 40013  df-covers 40100  df-ats 40101  df-atl 40132  df-cvlat 40156  df-hlat 40185  df-lhyp 40822

This theorem is used by: (None)