Theorem icoreclin 34632

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 34629	. . 3 ⊢ (𝑦 ∈ 𝐼 ↔ ∃𝑐 ∈ ℝ ∃𝑑 ∈ ℝ 𝑦 = {𝑧 ∈ ℝ ∣ (𝑐 ≤ 𝑧 ∧ 𝑧 < 𝑑)})
3	1	icoreelrnab 34629	. . . . . . 7 ⊢ (𝑥 ∈ 𝐼 ↔ ∃𝑎 ∈ ℝ ∃𝑏 ∈ ℝ 𝑥 = {𝑧 ∈ ℝ ∣ (𝑎 ≤ 𝑧 ∧ 𝑧 < 𝑏)})
4	1	isbasisrelowllem1 34630	. . . . . . . . . . . . 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 415	. . . . . . . . . . . 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 34631	. . . . . . . . . . . . 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 415	. . . . . . . . . . . 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 855	. . . . . . . . . . 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 4177	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∩ 𝑥) = (𝑥 ∩ 𝑦)
10	1	isbasisrelowllem2 34631	. . . . . . . . . . . . . . 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 2918	. . . . . . . . . . . . . 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 651	. . . . . . . . . . . . 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 415	. . . . . . . . . . . 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 34630	. . . . . . . . . . . . . . 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 2918	. . . . . . . . . . . . . 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 651	. . . . . . . . . . . . 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 415	. . . . . . . . . . . 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 855	. . . . . . . . . . 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 1144	. . . . . . . . . . . 12 ⊢ ((𝑎 ∈ ℝ ∧ 𝑏 ∈ ℝ ∧ 𝑥 = {𝑧 ∈ ℝ ∣ (𝑎 ≤ 𝑧 ∧ 𝑧 < 𝑏)}) → (𝑎 ∈ ℝ ∧ 𝑏 ∈ ℝ))
20		3simpa 1144	. . . . . . . . . . . 12 ⊢ ((𝑐 ∈ ℝ ∧ 𝑑 ∈ ℝ ∧ 𝑦 = {𝑧 ∈ ℝ ∣ (𝑐 ≤ 𝑧 ∧ 𝑧 < 𝑑)}) → (𝑐 ∈ ℝ ∧ 𝑑 ∈ ℝ))
21		letric 10734	. . . . . . . . . . . . . . 15 ⊢ ((𝑎 ∈ ℝ ∧ 𝑐 ∈ ℝ) → (𝑎 ≤ 𝑐 ∨ 𝑐 ≤ 𝑎))
22		letric 10734	. . . . . . . . . . . . . . 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 1007	. . . . . . . . . . . . . 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 220	. . . . . . . . . . . . 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 658	. . . . . . . . . . . 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 856	. . . . . . . . . 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 415	. . . . . . . . 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 1117	. . . . . . . 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 3292	. . . . . . 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 219	. . . . . 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 1117	. . . 4 ⊢ ((𝑐 ∈ ℝ ∧ 𝑑 ∈ ℝ) → (𝑦 = {𝑧 ∈ ℝ ∣ (𝑐 ≤ 𝑧 ∧ 𝑧 < 𝑑)} → (𝑥 ∈ 𝐼 → (𝑥 ∩ 𝑦) ∈ 𝐼)))
35	34	rexlimivv 3292	. . 3 ⊢ (∃𝑐 ∈ ℝ ∃𝑑 ∈ ℝ 𝑦 = {𝑧 ∈ ℝ ∣ (𝑐 ≤ 𝑧 ∧ 𝑧 < 𝑑)} → (𝑥 ∈ 𝐼 → (𝑥 ∩ 𝑦) ∈ 𝐼))
36	2, 35	sylbi 219	. 2 ⊢ (𝑦 ∈ 𝐼 → (𝑥 ∈ 𝐼 → (𝑥 ∩ 𝑦) ∈ 𝐼))
37	36	impcom 410	1 ⊢ ((𝑥 ∈ 𝐼 ∧ 𝑦 ∈ 𝐼) → (𝑥 ∩ 𝑦) ∈ 𝐼)

Syntax hints:   → wi 4   ∧ wa 398   ∨ wo 843   ∧ w3a 1083   = wceq 1533   ∈ wcel 2110  ∃wrex 3139  {crab 3142   ∩ cin 3934   class class class wbr 5058   × cxp 5547   “ cima 5552  ℝcr 10530   < clt 10669   ≤ cle 10670  [,)cico 12734

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-sep 5195  ax-nul 5202  ax-pow 5258  ax-pr 5321  ax-un 7455  ax-cnex 10587  ax-resscn 10588  ax-pre-lttri 10605  ax-pre-lttrn 10606

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-nel 3124  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4561  df-pr 4563  df-op 4567  df-uni 4832  df-iun 4913  df-br 5059  df-opab 5121  df-mpt 5139  df-id 5454  df-po 5468  df-so 5469  df-xp 5555  df-rel 5556  df-cnv 5557  df-co 5558  df-dm 5559  df-rn 5560  df-res 5561  df-ima 5562  df-iota 6308  df-fun 6351  df-fn 6352  df-f 6353  df-f1 6354  df-fo 6355  df-f1o 6356  df-fv 6357  df-ov 7153  df-oprab 7154  df-mpo 7155  df-1st 7683  df-2nd 7684  df-er 8283  df-en 8504  df-dom 8505  df-sdom 8506  df-pnf 10671  df-mnf 10672  df-xr 10673  df-ltxr 10674  df-le 10675  df-ico 12738

This theorem is referenced by:  isbasisrelowl  34633