Theorem lhprelat3N 40704

Description: The Hilbert lattice is relatively atomic with respect to co-atoms (lattice hyperplanes). Dual version of hlrelat3 40076. (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 489	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑋 < 𝑌) ∧ 𝑝 ∈ (Atoms‘𝐾)) → 𝑝 ∈ (Atoms‘𝐾))
2		simpll1 1229	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑋 < 𝑌) ∧ 𝑝 ∈ (Atoms‘𝐾)) → 𝐾 ∈ HL)
3		lhprelat3.b	. . . . . . . 8 ⊢ 𝐵 = (Base‘𝐾)
4		eqid 2769	. . . . . . . 8 ⊢ (Atoms‘𝐾) = (Atoms‘𝐾)
5	3, 4	atbase 39953	. . . . . . 7 ⊢ (𝑝 ∈ (Atoms‘𝐾) → 𝑝 ∈ 𝐵)
6	5	adantl 486	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑋 < 𝑌) ∧ 𝑝 ∈ (Atoms‘𝐾)) → 𝑝 ∈ 𝐵)
7		eqid 2769	. . . . . . 7 ⊢ (oc‘𝐾) = (oc‘𝐾)
8		lhprelat3.h	. . . . . . 7 ⊢ 𝐻 = (LHyp‘𝐾)
9	3, 7, 4, 8	lhpoc2N 40679	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ 𝑝 ∈ 𝐵) → (𝑝 ∈ (Atoms‘𝐾) ↔ ((oc‘𝐾)‘𝑝) ∈ 𝐻))
10	2, 6, 9	syl2anc 595	. . . . 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 485	. . 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 40026	. . . . . . . . 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 40028	. . . . . . . . 9 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑋 < 𝑌) ∧ 𝑝 ∈ (Atoms‘𝐾)) → 𝐾 ∈ Lat)
16		simpll3 1231	. . . . . . . . 9 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑋 < 𝑌) ∧ 𝑝 ∈ (Atoms‘𝐾)) → 𝑌 ∈ 𝐵)
17	3, 7	opoccl 39858	. . . . . . . . . 10 ⊢ ((𝐾 ∈ OP ∧ 𝑝 ∈ 𝐵) → ((oc‘𝐾)‘𝑝) ∈ 𝐵)
18	14, 6, 17	syl2anc 595	. . . . . . . . 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 18496	. . . . . . . . 9 ⊢ ((𝐾 ∈ Lat ∧ 𝑌 ∈ 𝐵 ∧ ((oc‘𝐾)‘𝑝) ∈ 𝐵) → (𝑌 ∧ ((oc‘𝐾)‘𝑝)) ∈ 𝐵)
21	15, 16, 18, 20	syl3anc 1396	. . . . . . . 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 39941	. . . . . . . 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 1396	. . . . . . 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 40025	. . . . . . . . . 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 2769	. . . . . . . . . 10 ⊢ (join‘𝐾) = (join‘𝐾)
28	3, 27, 19, 7	oldmm3N 39883	. . . . . . . . 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 1396	. . . . . . . 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 5125	. . . . . . 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 1230	. . . . . . . 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 39864	. . . . . . . 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 1396	. . . . . . 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 5123	. . . . . . 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 643	. . . . 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 481	. . . 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 466	. . 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 7419	. . . . . 6 ⊢ (𝑤 = ((oc‘𝐾)‘𝑝) → (𝑌 ∧ 𝑤) = (𝑌 ∧ ((oc‘𝐾)‘𝑝)))
42	41	breq2d 5125	. . . . 5 ⊢ (𝑤 = ((oc‘𝐾)‘𝑝) → (𝑋 ≤ (𝑌 ∧ 𝑤) ↔ 𝑋 ≤ (𝑌 ∧ ((oc‘𝐾)‘𝑝))))
43	41	breq1d 5123	. . . . 5 ⊢ (𝑤 = ((oc‘𝐾)‘𝑝) → ((𝑌 ∧ 𝑤)𝐶𝑌 ↔ (𝑌 ∧ ((oc‘𝐾)‘𝑝))𝐶𝑌))
44	42, 43	anbi12d 643	. . . 4 ⊢ (𝑤 = ((oc‘𝐾)‘𝑝) → ((𝑋 ≤ (𝑌 ∧ 𝑤) ∧ (𝑌 ∧ 𝑤)𝐶𝑌) ↔ (𝑋 ≤ (𝑌 ∧ ((oc‘𝐾)‘𝑝)) ∧ (𝑌 ∧ ((oc‘𝐾)‘𝑝))𝐶𝑌)))
45	44	rspcev 3590	. . 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 595	. 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 1208	. . 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 1210	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑋 < 𝑌) → 𝑌 ∈ 𝐵)
50	3, 7	opoccl 39858	. . . 4 ⊢ ((𝐾 ∈ OP ∧ 𝑌 ∈ 𝐵) → ((oc‘𝐾)‘𝑌) ∈ 𝐵)
51	48, 49, 50	syl2anc 595	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑋 < 𝑌) → ((oc‘𝐾)‘𝑌) ∈ 𝐵)
52		simpl2 1209	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑋 < 𝑌) → 𝑋 ∈ 𝐵)
53	3, 7	opoccl 39858	. . . 4 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵) → ((oc‘𝐾)‘𝑋) ∈ 𝐵)
54	48, 52, 53	syl2anc 595	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑋 < 𝑌) → ((oc‘𝐾)‘𝑋) ∈ 𝐵)
55		simpr 489	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑋 < 𝑌) → 𝑋 < 𝑌)
56		lhprelat3.s	. . . . . 6 ⊢ < = (lt‘𝐾)
57	3, 56, 7	opltcon3b 39868	. . . . 5 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 < 𝑌 ↔ ((oc‘𝐾)‘𝑌) < ((oc‘𝐾)‘𝑋)))
58	48, 52, 49, 57	syl3anc 1396	. . . 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 40076	. . 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 1398	. 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 3179	1 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑋 < 𝑌) → ∃𝑤 ∈ 𝐻 (𝑋 ≤ (𝑌 ∧ 𝑤) ∧ (𝑌 ∧ 𝑤)𝐶𝑌))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1101   = wceq 1567   ∈ wcel 2149  ∃wrex 3095   class class class wbr 5113  ‘cfv 6537  (class class class)co 7411  Basecbs 17269  lecple 17317  occoc 17318  ltcplt 18364  joincjn 18367  meetcmee 18368  Latclat 18487  OPcops 39836  OLcol 39838   ⋖ ccvr 39926  Atomscatm 39927  HLchlt 40014  LHypclh 40648

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-rep 5242  ax-sep 5261  ax-nul 5271  ax-pow 5337  ax-pr 5405  ax-un 7733

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-rmo 3376  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-iun 4962  df-br 5114  df-opab 5178  df-mpt 5197  df-id 5557  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7368  df-ov 7414  df-oprab 7415  df-proset 18350  df-poset 18369  df-plt 18384  df-lub 18400  df-glb 18401  df-join 18402  df-meet 18403  df-p0 18479  df-p1 18480  df-lat 18488  df-clat 18555  df-oposet 39840  df-ol 39842  df-oml 39843  df-covers 39930  df-ats 39931  df-atl 39962  df-cvlat 39986  df-hlat 40015  df-lhyp 40652

This theorem is referenced by: (None)