Theorem icoreclin 33205

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 33202	. . 3 ⊢ (𝑦 ∈ 𝐼 ↔ ∃𝑐 ∈ ℝ ∃𝑑 ∈ ℝ 𝑦 = {𝑧 ∈ ℝ ∣ (𝑐 ≤ 𝑧 ∧ 𝑧 < 𝑑)})
3	1	icoreelrnab 33202	. . . . . . 7 ⊢ (𝑥 ∈ 𝐼 ↔ ∃𝑎 ∈ ℝ ∃𝑏 ∈ ℝ 𝑥 = {𝑧 ∈ ℝ ∣ (𝑎 ≤ 𝑧 ∧ 𝑧 < 𝑏)})
4	1	isbasisrelowllem1 33203	. . . . . . . . . . . . 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 450	. . . . . . . . . . . 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 33204	. . . . . . . . . . . . 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 450	. . . . . . . . . . . 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 395	. . . . . . . . . . 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 3805	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∩ 𝑥) = (𝑥 ∩ 𝑦)
10	1	isbasisrelowllem2 33204	. . . . . . . . . . . . . . 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	syl5eqelr 2706	. . . . . . . . . . . . . 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 847	. . . . . . . . . . . . 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 450	. . . . . . . . . . . 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 33203	. . . . . . . . . . . . . . 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	syl5eqelr 2706	. . . . . . . . . . . . . 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 847	. . . . . . . . . . . . 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 450	. . . . . . . . . . . 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 395	. . . . . . . . . . 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 1058	. . . . . . . . . . . 12 ⊢ ((𝑎 ∈ ℝ ∧ 𝑏 ∈ ℝ ∧ 𝑥 = {𝑧 ∈ ℝ ∣ (𝑎 ≤ 𝑧 ∧ 𝑧 < 𝑏)}) → (𝑎 ∈ ℝ ∧ 𝑏 ∈ ℝ))
20		3simpa 1058	. . . . . . . . . . . 12 ⊢ ((𝑐 ∈ ℝ ∧ 𝑑 ∈ ℝ ∧ 𝑦 = {𝑧 ∈ ℝ ∣ (𝑐 ≤ 𝑧 ∧ 𝑧 < 𝑑)}) → (𝑐 ∈ ℝ ∧ 𝑑 ∈ ℝ))
21		letric 10137	. . . . . . . . . . . . . . 15 ⊢ ((𝑎 ∈ ℝ ∧ 𝑐 ∈ ℝ) → (𝑎 ≤ 𝑐 ∨ 𝑐 ≤ 𝑎))
22		letric 10137	. . . . . . . . . . . . . . 15 ⊢ ((𝑏 ∈ ℝ ∧ 𝑑 ∈ ℝ) → (𝑏 ≤ 𝑑 ∨ 𝑑 ≤ 𝑏))
23	21, 22	anim12i 590	. . . . . . . . . . . . . 14 ⊢ (((𝑎 ∈ ℝ ∧ 𝑐 ∈ ℝ) ∧ (𝑏 ∈ ℝ ∧ 𝑑 ∈ ℝ)) → ((𝑎 ≤ 𝑐 ∨ 𝑐 ≤ 𝑎) ∧ (𝑏 ≤ 𝑑 ∨ 𝑑 ≤ 𝑏)))
24		anddi 914	. . . . . . . . . . . . . 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 208	. . . . . . . . . . . . 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 869	. . . . . . . . . . . 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 494	. . . . . . . . . . 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 396	. . . . . . . . . 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 450	. . . . . . . . 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 1267	. . . . . . . 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 3036	. . . . . . 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 207	. . . . . 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 1267	. . . 4 ⊢ ((𝑐 ∈ ℝ ∧ 𝑑 ∈ ℝ) → (𝑦 = {𝑧 ∈ ℝ ∣ (𝑐 ≤ 𝑧 ∧ 𝑧 < 𝑑)} → (𝑥 ∈ 𝐼 → (𝑥 ∩ 𝑦) ∈ 𝐼)))
35	34	rexlimivv 3036	. . 3 ⊢ (∃𝑐 ∈ ℝ ∃𝑑 ∈ ℝ 𝑦 = {𝑧 ∈ ℝ ∣ (𝑐 ≤ 𝑧 ∧ 𝑧 < 𝑑)} → (𝑥 ∈ 𝐼 → (𝑥 ∩ 𝑦) ∈ 𝐼))
36	2, 35	sylbi 207	. 2 ⊢ (𝑦 ∈ 𝐼 → (𝑥 ∈ 𝐼 → (𝑥 ∩ 𝑦) ∈ 𝐼))
37	36	impcom 446	1 ⊢ ((𝑥 ∈ 𝐼 ∧ 𝑦 ∈ 𝐼) → (𝑥 ∩ 𝑦) ∈ 𝐼)

Syntax hints:   → wi 4   ∨ wo 383   ∧ wa 384   ∧ w3a 1037   = wceq 1483   ∈ wcel 1990  ∃wrex 2913  {crab 2916   ∩ cin 3573   class class class wbr 4653   × cxp 5112   “ cima 5117  ℝcr 9935   < clt 10074   ≤ cle 10075  [,)cico 12177

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-pre-lttri 10010  ax-pre-lttrn 10011

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-po 5035  df-so 5036  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-ico 12181

This theorem is referenced by:  isbasisrelowl  33206