Theorem lhprelat3N 37180

Description: The Hilbert lattice is relatively atomic with respect to co-atoms (lattice hyperplanes). Dual version of hlrelat3 36552. (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 487	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑋 < 𝑌) ∧ 𝑝 ∈ (Atoms‘𝐾)) → 𝑝 ∈ (Atoms‘𝐾))
2		simpll1 1208	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑋 < 𝑌) ∧ 𝑝 ∈ (Atoms‘𝐾)) → 𝐾 ∈ HL)
3		lhprelat3.b	. . . . . . . 8 ⊢ 𝐵 = (Base‘𝐾)
4		eqid 2824	. . . . . . . 8 ⊢ (Atoms‘𝐾) = (Atoms‘𝐾)
5	3, 4	atbase 36429	. . . . . . 7 ⊢ (𝑝 ∈ (Atoms‘𝐾) → 𝑝 ∈ 𝐵)
6	5	adantl 484	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑋 < 𝑌) ∧ 𝑝 ∈ (Atoms‘𝐾)) → 𝑝 ∈ 𝐵)
7		eqid 2824	. . . . . . 7 ⊢ (oc‘𝐾) = (oc‘𝐾)
8		lhprelat3.h	. . . . . . 7 ⊢ 𝐻 = (LHyp‘𝐾)
9	3, 7, 4, 8	lhpoc2N 37155	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ 𝑝 ∈ 𝐵) → (𝑝 ∈ (Atoms‘𝐾) ↔ ((oc‘𝐾)‘𝑝) ∈ 𝐻))
10	2, 6, 9	syl2anc 586	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑋 < 𝑌) ∧ 𝑝 ∈ (Atoms‘𝐾)) → (𝑝 ∈ (Atoms‘𝐾) ↔ ((oc‘𝐾)‘𝑝) ∈ 𝐻))
11	1, 10	mpbid 234	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑋 < 𝑌) ∧ 𝑝 ∈ (Atoms‘𝐾)) → ((oc‘𝐾)‘𝑝) ∈ 𝐻)
12	11	adantr 483	. . 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 36502	. . . . . . . . 9 ⊢ (𝐾 ∈ HL → 𝐾 ∈ OP)
14	2, 13	syl 17	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑋 < 𝑌) ∧ 𝑝 ∈ (Atoms‘𝐾)) → 𝐾 ∈ OP)
15	2	hllatd 36504	. . . . . . . . 9 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑋 < 𝑌) ∧ 𝑝 ∈ (Atoms‘𝐾)) → 𝐾 ∈ Lat)
16		simpll3 1210	. . . . . . . . 9 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑋 < 𝑌) ∧ 𝑝 ∈ (Atoms‘𝐾)) → 𝑌 ∈ 𝐵)
17	3, 7	opoccl 36334	. . . . . . . . . 10 ⊢ ((𝐾 ∈ OP ∧ 𝑝 ∈ 𝐵) → ((oc‘𝐾)‘𝑝) ∈ 𝐵)
18	14, 6, 17	syl2anc 586	. . . . . . . . 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 17665	. . . . . . . . 9 ⊢ ((𝐾 ∈ Lat ∧ 𝑌 ∈ 𝐵 ∧ ((oc‘𝐾)‘𝑝) ∈ 𝐵) → (𝑌 ∧ ((oc‘𝐾)‘𝑝)) ∈ 𝐵)
21	15, 16, 18, 20	syl3anc 1367	. . . . . . . 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 36417	. . . . . . . 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 1367	. . . . . . 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 36501	. . . . . . . . . 10 ⊢ (𝐾 ∈ HL → 𝐾 ∈ OL)
26	2, 25	syl 17	. . . . . . . . 9 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑋 < 𝑌) ∧ 𝑝 ∈ (Atoms‘𝐾)) → 𝐾 ∈ OL)
27		eqid 2824	. . . . . . . . . 10 ⊢ (join‘𝐾) = (join‘𝐾)
28	3, 27, 19, 7	oldmm3N 36359	. . . . . . . . 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 1367	. . . . . . . 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 5081	. . . . . . 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 282	. . . . . 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 1209	. . . . . . . 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 36340	. . . . . . . 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 1367	. . . . . . 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 5079	. . . . . . 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 282	. . . . . 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 632	. . . . 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 479	. . . 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 464	. . 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 7167	. . . . . 6 ⊢ (𝑤 = ((oc‘𝐾)‘𝑝) → (𝑌 ∧ 𝑤) = (𝑌 ∧ ((oc‘𝐾)‘𝑝)))
42	41	breq2d 5081	. . . . 5 ⊢ (𝑤 = ((oc‘𝐾)‘𝑝) → (𝑋 ≤ (𝑌 ∧ 𝑤) ↔ 𝑋 ≤ (𝑌 ∧ ((oc‘𝐾)‘𝑝))))
43	41	breq1d 5079	. . . . 5 ⊢ (𝑤 = ((oc‘𝐾)‘𝑝) → ((𝑌 ∧ 𝑤)𝐶𝑌 ↔ (𝑌 ∧ ((oc‘𝐾)‘𝑝))𝐶𝑌))
44	42, 43	anbi12d 632	. . . 4 ⊢ (𝑤 = ((oc‘𝐾)‘𝑝) → ((𝑋 ≤ (𝑌 ∧ 𝑤) ∧ (𝑌 ∧ 𝑤)𝐶𝑌) ↔ (𝑋 ≤ (𝑌 ∧ ((oc‘𝐾)‘𝑝)) ∧ (𝑌 ∧ ((oc‘𝐾)‘𝑝))𝐶𝑌)))
45	44	rspcev 3626	. . 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 586	. 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 1187	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑋 < 𝑌) → 𝐾 ∈ HL)
48	47, 13	syl 17	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑋 < 𝑌) → 𝐾 ∈ OP)
49		simpl3 1189	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑋 < 𝑌) → 𝑌 ∈ 𝐵)
50	3, 7	opoccl 36334	. . . 4 ⊢ ((𝐾 ∈ OP ∧ 𝑌 ∈ 𝐵) → ((oc‘𝐾)‘𝑌) ∈ 𝐵)
51	48, 49, 50	syl2anc 586	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑋 < 𝑌) → ((oc‘𝐾)‘𝑌) ∈ 𝐵)
52		simpl2 1188	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑋 < 𝑌) → 𝑋 ∈ 𝐵)
53	3, 7	opoccl 36334	. . . 4 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵) → ((oc‘𝐾)‘𝑋) ∈ 𝐵)
54	48, 52, 53	syl2anc 586	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑋 < 𝑌) → ((oc‘𝐾)‘𝑋) ∈ 𝐵)
55		simpr 487	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑋 < 𝑌) → 𝑋 < 𝑌)
56		lhprelat3.s	. . . . . 6 ⊢ < = (lt‘𝐾)
57	3, 56, 7	opltcon3b 36344	. . . . 5 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 < 𝑌 ↔ ((oc‘𝐾)‘𝑌) < ((oc‘𝐾)‘𝑋)))
58	48, 52, 49, 57	syl3anc 1367	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑋 < 𝑌) → (𝑋 < 𝑌 ↔ ((oc‘𝐾)‘𝑌) < ((oc‘𝐾)‘𝑋)))
59	55, 58	mpbid 234	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑋 < 𝑌) → ((oc‘𝐾)‘𝑌) < ((oc‘𝐾)‘𝑋))
60	3, 33, 56, 27, 22, 4	hlrelat3 36552	. . 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 1369	. 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 3292	1 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑋 < 𝑌) → ∃𝑤 ∈ 𝐻 (𝑋 ≤ (𝑌 ∧ 𝑤) ∧ (𝑌 ∧ 𝑤)𝐶𝑌))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   ∧ w3a 1083   = wceq 1536   ∈ wcel 2113  ∃wrex 3142   class class class wbr 5069  ‘cfv 6358  (class class class)co 7159  Basecbs 16486  lecple 16575  occoc 16576  ltcplt 17554  joincjn 17557  meetcmee 17558  Latclat 17658  OPcops 36312  OLcol 36314   ⋖ ccvr 36402  Atomscatm 36403  HLchlt 36490  LHypclh 37124

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2796  ax-rep 5193  ax-sep 5206  ax-nul 5213  ax-pow 5269  ax-pr 5333  ax-un 7464

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-mo 2621  df-eu 2653  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2966  df-ne 3020  df-ral 3146  df-rex 3147  df-reu 3148  df-rab 3150  df-v 3499  df-sbc 3776  df-csb 3887  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4471  df-pw 4544  df-sn 4571  df-pr 4573  df-op 4577  df-uni 4842  df-iun 4924  df-br 5070  df-opab 5132  df-mpt 5150  df-id 5463  df-xp 5564  df-rel 5565  df-cnv 5566  df-co 5567  df-dm 5568  df-rn 5569  df-res 5570  df-ima 5571  df-iota 6317  df-fun 6360  df-fn 6361  df-f 6362  df-f1 6363  df-fo 6364  df-f1o 6365  df-fv 6366  df-riota 7117  df-ov 7162  df-oprab 7163  df-proset 17541  df-poset 17559  df-plt 17571  df-lub 17587  df-glb 17588  df-join 17589  df-meet 17590  df-p0 17652  df-p1 17653  df-lat 17659  df-clat 17721  df-oposet 36316  df-ol 36318  df-oml 36319  df-covers 36406  df-ats 36407  df-atl 36438  df-cvlat 36462  df-hlat 36491  df-lhyp 37128

This theorem is referenced by: (None)