Theorem icoreclin 36227

Description: The set of closed-below, open-above intervals of reals is closed under finite intersection. (Contributed by ML, 27-Jul-2020.)

Hypothesis

Ref	Expression
isbasisrelowl.1	⊢ 𝐼 = ([,) “ (ℝ × ℝ))

Assertion

Ref	Expression
icoreclin	⊢ ((𝑥 ∈ 𝐼 ∧ 𝑦 ∈ 𝐼) → (𝑥 ∩ 𝑦) ∈ 𝐼)

Distinct variable group:   𝑥,𝐼,𝑦

Proof of Theorem icoreclin

Dummy variables 𝑧 𝑎 𝑏 𝑐 𝑑 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		isbasisrelowl.1	. . . 4 ⊢ 𝐼 = ([,) “ (ℝ × ℝ))
2	1	icoreelrnab 36224	. . 3 ⊢ (𝑦 ∈ 𝐼 ↔ ∃𝑐 ∈ ℝ ∃𝑑 ∈ ℝ 𝑦 = {𝑧 ∈ ℝ ∣ (𝑐 ≤ 𝑧 ∧ 𝑧 < 𝑑)})
3	1	icoreelrnab 36224	. . . . . . 7 ⊢ (𝑥 ∈ 𝐼 ↔ ∃𝑎 ∈ ℝ ∃𝑏 ∈ ℝ 𝑥 = {𝑧 ∈ ℝ ∣ (𝑎 ≤ 𝑧 ∧ 𝑧 < 𝑏)})
4	1	isbasisrelowllem1 36225	. . . . . . . . . . . . 13 ⊢ ((((𝑎 ∈ ℝ ∧ 𝑏 ∈ ℝ ∧ 𝑥 = {𝑧 ∈ ℝ ∣ (𝑎 ≤ 𝑧 ∧ 𝑧 < 𝑏)}) ∧ (𝑐 ∈ ℝ ∧ 𝑑 ∈ ℝ ∧ 𝑦 = {𝑧 ∈ ℝ ∣ (𝑐 ≤ 𝑧 ∧ 𝑧 < 𝑑)})) ∧ (𝑎 ≤ 𝑐 ∧ 𝑏 ≤ 𝑑)) → (𝑥 ∩ 𝑦) ∈ 𝐼)
5	4	ex 414	. . . . . . . . . . . 12 ⊢ (((𝑎 ∈ ℝ ∧ 𝑏 ∈ ℝ ∧ 𝑥 = {𝑧 ∈ ℝ ∣ (𝑎 ≤ 𝑧 ∧ 𝑧 < 𝑏)}) ∧ (𝑐 ∈ ℝ ∧ 𝑑 ∈ ℝ ∧ 𝑦 = {𝑧 ∈ ℝ ∣ (𝑐 ≤ 𝑧 ∧ 𝑧 < 𝑑)})) → ((𝑎 ≤ 𝑐 ∧ 𝑏 ≤ 𝑑) → (𝑥 ∩ 𝑦) ∈ 𝐼))
6	1	isbasisrelowllem2 36226	. . . . . . . . . . . . 13 ⊢ ((((𝑎 ∈ ℝ ∧ 𝑏 ∈ ℝ ∧ 𝑥 = {𝑧 ∈ ℝ ∣ (𝑎 ≤ 𝑧 ∧ 𝑧 < 𝑏)}) ∧ (𝑐 ∈ ℝ ∧ 𝑑 ∈ ℝ ∧ 𝑦 = {𝑧 ∈ ℝ ∣ (𝑐 ≤ 𝑧 ∧ 𝑧 < 𝑑)})) ∧ (𝑎 ≤ 𝑐 ∧ 𝑑 ≤ 𝑏)) → (𝑥 ∩ 𝑦) ∈ 𝐼)
7	6	ex 414	. . . . . . . . . . . 12 ⊢ (((𝑎 ∈ ℝ ∧ 𝑏 ∈ ℝ ∧ 𝑥 = {𝑧 ∈ ℝ ∣ (𝑎 ≤ 𝑧 ∧ 𝑧 < 𝑏)}) ∧ (𝑐 ∈ ℝ ∧ 𝑑 ∈ ℝ ∧ 𝑦 = {𝑧 ∈ ℝ ∣ (𝑐 ≤ 𝑧 ∧ 𝑧 < 𝑑)})) → ((𝑎 ≤ 𝑐 ∧ 𝑑 ≤ 𝑏) → (𝑥 ∩ 𝑦) ∈ 𝐼))
8	5, 7	jaod 858	. . . . . . . . . . 11 ⊢ (((𝑎 ∈ ℝ ∧ 𝑏 ∈ ℝ ∧ 𝑥 = {𝑧 ∈ ℝ ∣ (𝑎 ≤ 𝑧 ∧ 𝑧 < 𝑏)}) ∧ (𝑐 ∈ ℝ ∧ 𝑑 ∈ ℝ ∧ 𝑦 = {𝑧 ∈ ℝ ∣ (𝑐 ≤ 𝑧 ∧ 𝑧 < 𝑑)})) → (((𝑎 ≤ 𝑐 ∧ 𝑏 ≤ 𝑑) ∨ (𝑎 ≤ 𝑐 ∧ 𝑑 ≤ 𝑏)) → (𝑥 ∩ 𝑦) ∈ 𝐼))
9		incom 4201	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∩ 𝑥) = (𝑥 ∩ 𝑦)
10	1	isbasisrelowllem2 36226	. . . . . . . . . . . . . . 15 ⊢ ((((𝑐 ∈ ℝ ∧ 𝑑 ∈ ℝ ∧ 𝑦 = {𝑧 ∈ ℝ ∣ (𝑐 ≤ 𝑧 ∧ 𝑧 < 𝑑)}) ∧ (𝑎 ∈ ℝ ∧ 𝑏 ∈ ℝ ∧ 𝑥 = {𝑧 ∈ ℝ ∣ (𝑎 ≤ 𝑧 ∧ 𝑧 < 𝑏)})) ∧ (𝑐 ≤ 𝑎 ∧ 𝑏 ≤ 𝑑)) → (𝑦 ∩ 𝑥) ∈ 𝐼)
11	9, 10	eqeltrrid 2839	. . . . . . . . . . . . . 14 ⊢ ((((𝑐 ∈ ℝ ∧ 𝑑 ∈ ℝ ∧ 𝑦 = {𝑧 ∈ ℝ ∣ (𝑐 ≤ 𝑧 ∧ 𝑧 < 𝑑)}) ∧ (𝑎 ∈ ℝ ∧ 𝑏 ∈ ℝ ∧ 𝑥 = {𝑧 ∈ ℝ ∣ (𝑎 ≤ 𝑧 ∧ 𝑧 < 𝑏)})) ∧ (𝑐 ≤ 𝑎 ∧ 𝑏 ≤ 𝑑)) → (𝑥 ∩ 𝑦) ∈ 𝐼)
12	11	ancom1s 652	. . . . . . . . . . . . 13 ⊢ ((((𝑎 ∈ ℝ ∧ 𝑏 ∈ ℝ ∧ 𝑥 = {𝑧 ∈ ℝ ∣ (𝑎 ≤ 𝑧 ∧ 𝑧 < 𝑏)}) ∧ (𝑐 ∈ ℝ ∧ 𝑑 ∈ ℝ ∧ 𝑦 = {𝑧 ∈ ℝ ∣ (𝑐 ≤ 𝑧 ∧ 𝑧 < 𝑑)})) ∧ (𝑐 ≤ 𝑎 ∧ 𝑏 ≤ 𝑑)) → (𝑥 ∩ 𝑦) ∈ 𝐼)
13	12	ex 414	. . . . . . . . . . . 12 ⊢ (((𝑎 ∈ ℝ ∧ 𝑏 ∈ ℝ ∧ 𝑥 = {𝑧 ∈ ℝ ∣ (𝑎 ≤ 𝑧 ∧ 𝑧 < 𝑏)}) ∧ (𝑐 ∈ ℝ ∧ 𝑑 ∈ ℝ ∧ 𝑦 = {𝑧 ∈ ℝ ∣ (𝑐 ≤ 𝑧 ∧ 𝑧 < 𝑑)})) → ((𝑐 ≤ 𝑎 ∧ 𝑏 ≤ 𝑑) → (𝑥 ∩ 𝑦) ∈ 𝐼))
14	1	isbasisrelowllem1 36225	. . . . . . . . . . . . . . 15 ⊢ ((((𝑐 ∈ ℝ ∧ 𝑑 ∈ ℝ ∧ 𝑦 = {𝑧 ∈ ℝ ∣ (𝑐 ≤ 𝑧 ∧ 𝑧 < 𝑑)}) ∧ (𝑎 ∈ ℝ ∧ 𝑏 ∈ ℝ ∧ 𝑥 = {𝑧 ∈ ℝ ∣ (𝑎 ≤ 𝑧 ∧ 𝑧 < 𝑏)})) ∧ (𝑐 ≤ 𝑎 ∧ 𝑑 ≤ 𝑏)) → (𝑦 ∩ 𝑥) ∈ 𝐼)
15	9, 14	eqeltrrid 2839	. . . . . . . . . . . . . 14 ⊢ ((((𝑐 ∈ ℝ ∧ 𝑑 ∈ ℝ ∧ 𝑦 = {𝑧 ∈ ℝ ∣ (𝑐 ≤ 𝑧 ∧ 𝑧 < 𝑑)}) ∧ (𝑎 ∈ ℝ ∧ 𝑏 ∈ ℝ ∧ 𝑥 = {𝑧 ∈ ℝ ∣ (𝑎 ≤ 𝑧 ∧ 𝑧 < 𝑏)})) ∧ (𝑐 ≤ 𝑎 ∧ 𝑑 ≤ 𝑏)) → (𝑥 ∩ 𝑦) ∈ 𝐼)
16	15	ancom1s 652	. . . . . . . . . . . . 13 ⊢ ((((𝑎 ∈ ℝ ∧ 𝑏 ∈ ℝ ∧ 𝑥 = {𝑧 ∈ ℝ ∣ (𝑎 ≤ 𝑧 ∧ 𝑧 < 𝑏)}) ∧ (𝑐 ∈ ℝ ∧ 𝑑 ∈ ℝ ∧ 𝑦 = {𝑧 ∈ ℝ ∣ (𝑐 ≤ 𝑧 ∧ 𝑧 < 𝑑)})) ∧ (𝑐 ≤ 𝑎 ∧ 𝑑 ≤ 𝑏)) → (𝑥 ∩ 𝑦) ∈ 𝐼)
17	16	ex 414	. . . . . . . . . . . 12 ⊢ (((𝑎 ∈ ℝ ∧ 𝑏 ∈ ℝ ∧ 𝑥 = {𝑧 ∈ ℝ ∣ (𝑎 ≤ 𝑧 ∧ 𝑧 < 𝑏)}) ∧ (𝑐 ∈ ℝ ∧ 𝑑 ∈ ℝ ∧ 𝑦 = {𝑧 ∈ ℝ ∣ (𝑐 ≤ 𝑧 ∧ 𝑧 < 𝑑)})) → ((𝑐 ≤ 𝑎 ∧ 𝑑 ≤ 𝑏) → (𝑥 ∩ 𝑦) ∈ 𝐼))
18	13, 17	jaod 858	. . . . . . . . . . 11 ⊢ (((𝑎 ∈ ℝ ∧ 𝑏 ∈ ℝ ∧ 𝑥 = {𝑧 ∈ ℝ ∣ (𝑎 ≤ 𝑧 ∧ 𝑧 < 𝑏)}) ∧ (𝑐 ∈ ℝ ∧ 𝑑 ∈ ℝ ∧ 𝑦 = {𝑧 ∈ ℝ ∣ (𝑐 ≤ 𝑧 ∧ 𝑧 < 𝑑)})) → (((𝑐 ≤ 𝑎 ∧ 𝑏 ≤ 𝑑) ∨ (𝑐 ≤ 𝑎 ∧ 𝑑 ≤ 𝑏)) → (𝑥 ∩ 𝑦) ∈ 𝐼))
19		3simpa 1149	. . . . . . . . . . . 12 ⊢ ((𝑎 ∈ ℝ ∧ 𝑏 ∈ ℝ ∧ 𝑥 = {𝑧 ∈ ℝ ∣ (𝑎 ≤ 𝑧 ∧ 𝑧 < 𝑏)}) → (𝑎 ∈ ℝ ∧ 𝑏 ∈ ℝ))
20		3simpa 1149	. . . . . . . . . . . 12 ⊢ ((𝑐 ∈ ℝ ∧ 𝑑 ∈ ℝ ∧ 𝑦 = {𝑧 ∈ ℝ ∣ (𝑐 ≤ 𝑧 ∧ 𝑧 < 𝑑)}) → (𝑐 ∈ ℝ ∧ 𝑑 ∈ ℝ))
21		letric 11311	. . . . . . . . . . . . . . 15 ⊢ ((𝑎 ∈ ℝ ∧ 𝑐 ∈ ℝ) → (𝑎 ≤ 𝑐 ∨ 𝑐 ≤ 𝑎))
22		letric 11311	. . . . . . . . . . . . . . 15 ⊢ ((𝑏 ∈ ℝ ∧ 𝑑 ∈ ℝ) → (𝑏 ≤ 𝑑 ∨ 𝑑 ≤ 𝑏))
23	21, 22	anim12i 614	. . . . . . . . . . . . . 14 ⊢ (((𝑎 ∈ ℝ ∧ 𝑐 ∈ ℝ) ∧ (𝑏 ∈ ℝ ∧ 𝑑 ∈ ℝ)) → ((𝑎 ≤ 𝑐 ∨ 𝑐 ≤ 𝑎) ∧ (𝑏 ≤ 𝑑 ∨ 𝑑 ≤ 𝑏)))
24		anddi 1010	. . . . . . . . . . . . . 14 ⊢ (((𝑎 ≤ 𝑐 ∨ 𝑐 ≤ 𝑎) ∧ (𝑏 ≤ 𝑑 ∨ 𝑑 ≤ 𝑏)) ↔ (((𝑎 ≤ 𝑐 ∧ 𝑏 ≤ 𝑑) ∨ (𝑎 ≤ 𝑐 ∧ 𝑑 ≤ 𝑏)) ∨ ((𝑐 ≤ 𝑎 ∧ 𝑏 ≤ 𝑑) ∨ (𝑐 ≤ 𝑎 ∧ 𝑑 ≤ 𝑏))))
25	23, 24	sylib 217	. . . . . . . . . . . . 13 ⊢ (((𝑎 ∈ ℝ ∧ 𝑐 ∈ ℝ) ∧ (𝑏 ∈ ℝ ∧ 𝑑 ∈ ℝ)) → (((𝑎 ≤ 𝑐 ∧ 𝑏 ≤ 𝑑) ∨ (𝑎 ≤ 𝑐 ∧ 𝑑 ≤ 𝑏)) ∨ ((𝑐 ≤ 𝑎 ∧ 𝑏 ≤ 𝑑) ∨ (𝑐 ≤ 𝑎 ∧ 𝑑 ≤ 𝑏))))
26	25	an4s 659	. . . . . . . . . . . 12 ⊢ (((𝑎 ∈ ℝ ∧ 𝑏 ∈ ℝ) ∧ (𝑐 ∈ ℝ ∧ 𝑑 ∈ ℝ)) → (((𝑎 ≤ 𝑐 ∧ 𝑏 ≤ 𝑑) ∨ (𝑎 ≤ 𝑐 ∧ 𝑑 ≤ 𝑏)) ∨ ((𝑐 ≤ 𝑎 ∧ 𝑏 ≤ 𝑑) ∨ (𝑐 ≤ 𝑎 ∧ 𝑑 ≤ 𝑏))))
27	19, 20, 26	syl2an 597	. . . . . . . . . . 11 ⊢ (((𝑎 ∈ ℝ ∧ 𝑏 ∈ ℝ ∧ 𝑥 = {𝑧 ∈ ℝ ∣ (𝑎 ≤ 𝑧 ∧ 𝑧 < 𝑏)}) ∧ (𝑐 ∈ ℝ ∧ 𝑑 ∈ ℝ ∧ 𝑦 = {𝑧 ∈ ℝ ∣ (𝑐 ≤ 𝑧 ∧ 𝑧 < 𝑑)})) → (((𝑎 ≤ 𝑐 ∧ 𝑏 ≤ 𝑑) ∨ (𝑎 ≤ 𝑐 ∧ 𝑑 ≤ 𝑏)) ∨ ((𝑐 ≤ 𝑎 ∧ 𝑏 ≤ 𝑑) ∨ (𝑐 ≤ 𝑎 ∧ 𝑑 ≤ 𝑏))))
28	8, 18, 27	mpjaod 859	. . . . . . . . . 10 ⊢ (((𝑎 ∈ ℝ ∧ 𝑏 ∈ ℝ ∧ 𝑥 = {𝑧 ∈ ℝ ∣ (𝑎 ≤ 𝑧 ∧ 𝑧 < 𝑏)}) ∧ (𝑐 ∈ ℝ ∧ 𝑑 ∈ ℝ ∧ 𝑦 = {𝑧 ∈ ℝ ∣ (𝑐 ≤ 𝑧 ∧ 𝑧 < 𝑑)})) → (𝑥 ∩ 𝑦) ∈ 𝐼)
29	28	ex 414	. . . . . . . . 9 ⊢ ((𝑎 ∈ ℝ ∧ 𝑏 ∈ ℝ ∧ 𝑥 = {𝑧 ∈ ℝ ∣ (𝑎 ≤ 𝑧 ∧ 𝑧 < 𝑏)}) → ((𝑐 ∈ ℝ ∧ 𝑑 ∈ ℝ ∧ 𝑦 = {𝑧 ∈ ℝ ∣ (𝑐 ≤ 𝑧 ∧ 𝑧 < 𝑑)}) → (𝑥 ∩ 𝑦) ∈ 𝐼))
30	29	3expia 1122	. . . . . . . 8 ⊢ ((𝑎 ∈ ℝ ∧ 𝑏 ∈ ℝ) → (𝑥 = {𝑧 ∈ ℝ ∣ (𝑎 ≤ 𝑧 ∧ 𝑧 < 𝑏)} → ((𝑐 ∈ ℝ ∧ 𝑑 ∈ ℝ ∧ 𝑦 = {𝑧 ∈ ℝ ∣ (𝑐 ≤ 𝑧 ∧ 𝑧 < 𝑑)}) → (𝑥 ∩ 𝑦) ∈ 𝐼)))
31	30	rexlimivv 3200	. . . . . . 7 ⊢ (∃𝑎 ∈ ℝ ∃𝑏 ∈ ℝ 𝑥 = {𝑧 ∈ ℝ ∣ (𝑎 ≤ 𝑧 ∧ 𝑧 < 𝑏)} → ((𝑐 ∈ ℝ ∧ 𝑑 ∈ ℝ ∧ 𝑦 = {𝑧 ∈ ℝ ∣ (𝑐 ≤ 𝑧 ∧ 𝑧 < 𝑑)}) → (𝑥 ∩ 𝑦) ∈ 𝐼))
32	3, 31	sylbi 216	. . . . . 6 ⊢ (𝑥 ∈ 𝐼 → ((𝑐 ∈ ℝ ∧ 𝑑 ∈ ℝ ∧ 𝑦 = {𝑧 ∈ ℝ ∣ (𝑐 ≤ 𝑧 ∧ 𝑧 < 𝑑)}) → (𝑥 ∩ 𝑦) ∈ 𝐼))
33	32	com12 32	. . . . 5 ⊢ ((𝑐 ∈ ℝ ∧ 𝑑 ∈ ℝ ∧ 𝑦 = {𝑧 ∈ ℝ ∣ (𝑐 ≤ 𝑧 ∧ 𝑧 < 𝑑)}) → (𝑥 ∈ 𝐼 → (𝑥 ∩ 𝑦) ∈ 𝐼))
34	33	3expia 1122	. . . 4 ⊢ ((𝑐 ∈ ℝ ∧ 𝑑 ∈ ℝ) → (𝑦 = {𝑧 ∈ ℝ ∣ (𝑐 ≤ 𝑧 ∧ 𝑧 < 𝑑)} → (𝑥 ∈ 𝐼 → (𝑥 ∩ 𝑦) ∈ 𝐼)))
35	34	rexlimivv 3200	. . 3 ⊢ (∃𝑐 ∈ ℝ ∃𝑑 ∈ ℝ 𝑦 = {𝑧 ∈ ℝ ∣ (𝑐 ≤ 𝑧 ∧ 𝑧 < 𝑑)} → (𝑥 ∈ 𝐼 → (𝑥 ∩ 𝑦) ∈ 𝐼))
36	2, 35	sylbi 216	. 2 ⊢ (𝑦 ∈ 𝐼 → (𝑥 ∈ 𝐼 → (𝑥 ∩ 𝑦) ∈ 𝐼))
37	36	impcom 409	1 ⊢ ((𝑥 ∈ 𝐼 ∧ 𝑦 ∈ 𝐼) → (𝑥 ∩ 𝑦) ∈ 𝐼)

Syntax hints:   → wi 4   ∧ wa 397   ∨ wo 846   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107  ∃wrex 3071  {crab 3433   ∩ cin 3947   class class class wbr 5148   × cxp 5674   “ cima 5679  ℝcr 11106   < clt 11245   ≤ cle 11246  [,)cico 13323

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7722  ax-cnex 11163  ax-resscn 11164  ax-pre-lttri 11181  ax-pre-lttrn 11182

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-po 5588  df-so 5589  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6493  df-fun 6543  df-fn 6544  df-f 6545  df-f1 6546  df-fo 6547  df-f1o 6548  df-fv 6549  df-ov 7409  df-oprab 7410  df-mpo 7411  df-1st 7972  df-2nd 7973  df-er 8700  df-en 8937  df-dom 8938  df-sdom 8939  df-pnf 11247  df-mnf 11248  df-xr 11249  df-ltxr 11250  df-le 11251  df-ico 13327

This theorem is referenced by:  isbasisrelowl  36228