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Theorem dedekindicclemlu 14147

Description: Lemma for dedekindicc 14150. There is a number which separates the lower and upper cuts. (Contributed by Jim Kingdon, 15-Feb-2024.)

**Hypotheses**
Ref	Expression
dedekindicc.a	⊢ (𝜑 → 𝐴 ∈ ℝ)
dedekindicc.b	⊢ (𝜑 → 𝐵 ∈ ℝ)
dedekindicc.lss	⊢ (𝜑 → 𝐿 ⊆ (𝐴[,]𝐵))
dedekindicc.uss	⊢ (𝜑 → 𝑈 ⊆ (𝐴[,]𝐵))
dedekindicc.lm	⊢ (𝜑 → ∃𝑞 ∈ (𝐴[,]𝐵)𝑞 ∈ 𝐿)
dedekindicc.um	⊢ (𝜑 → ∃𝑟 ∈ (𝐴[,]𝐵)𝑟 ∈ 𝑈)
dedekindicc.lr	⊢ (𝜑 → ∀𝑞 ∈ (𝐴[,]𝐵)(𝑞 ∈ 𝐿 ↔ ∃𝑟 ∈ 𝐿 𝑞 < 𝑟))
dedekindicc.ur	⊢ (𝜑 → ∀𝑟 ∈ (𝐴[,]𝐵)(𝑟 ∈ 𝑈 ↔ ∃𝑞 ∈ 𝑈 𝑞 < 𝑟))
dedekindicc.disj	⊢ (𝜑 → (𝐿 ∩ 𝑈) = ∅)
dedekindicc.loc	⊢ (𝜑 → ∀𝑞 ∈ (𝐴[,]𝐵)∀𝑟 ∈ (𝐴[,]𝐵)(𝑞 < 𝑟 → (𝑞 ∈ 𝐿 ∨ 𝑟 ∈ 𝑈)))
dedekindicc.ab	⊢ (𝜑 → 𝐴 < 𝐵)

**Assertion**
Ref	Expression
dedekindicclemlu	⊢ (𝜑 → ∃𝑥 ∈ (𝐴[,]𝐵)(∀𝑞 ∈ 𝐿 𝑞 < 𝑥 ∧ ∀𝑟 ∈ 𝑈 𝑥 < 𝑟))

Distinct variable groups: 𝐴,𝑞,𝑟,𝑥 𝐵,𝑞,𝑟,𝑥 𝐿,𝑞,𝑟,𝑥 𝑈,𝑞,𝑟 𝜑,𝑞,𝑟,𝑥 Allowed substitution hint: 𝑈(𝑥)

Proof of Theorem dedekindicclemlu Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		dedekindicc.a	. . 3 ⊢ (𝜑 → 𝐴 ∈ ℝ)
2		dedekindicc.b	. . 3 ⊢ (𝜑 → 𝐵 ∈ ℝ)
3		dedekindicc.lss	. . 3 ⊢ (𝜑 → 𝐿 ⊆ (𝐴[,]𝐵))
4		dedekindicc.uss	. . 3 ⊢ (𝜑 → 𝑈 ⊆ (𝐴[,]𝐵))
5		dedekindicc.lm	. . 3 ⊢ (𝜑 → ∃𝑞 ∈ (𝐴[,]𝐵)𝑞 ∈ 𝐿)
6		dedekindicc.um	. . 3 ⊢ (𝜑 → ∃𝑟 ∈ (𝐴[,]𝐵)𝑟 ∈ 𝑈)
7		dedekindicc.lr	. . 3 ⊢ (𝜑 → ∀𝑞 ∈ (𝐴[,]𝐵)(𝑞 ∈ 𝐿 ↔ ∃𝑟 ∈ 𝐿 𝑞 < 𝑟))
8		dedekindicc.ur	. . 3 ⊢ (𝜑 → ∀𝑟 ∈ (𝐴[,]𝐵)(𝑟 ∈ 𝑈 ↔ ∃𝑞 ∈ 𝑈 𝑞 < 𝑟))
9		dedekindicc.disj	. . 3 ⊢ (𝜑 → (𝐿 ∩ 𝑈) = ∅)
10		dedekindicc.loc	. . 3 ⊢ (𝜑 → ∀𝑞 ∈ (𝐴[,]𝐵)∀𝑟 ∈ (𝐴[,]𝐵)(𝑞 < 𝑟 → (𝑞 ∈ 𝐿 ∨ 𝑟 ∈ 𝑈)))
11		dedekindicc.ab	. . 3 ⊢ (𝜑 → 𝐴 < 𝐵)
12	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11	dedekindicclemlub 14146	. 2 ⊢ (𝜑 → ∃𝑥 ∈ (𝐴[,]𝐵)(∀𝑦 ∈ 𝐿 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ (𝐴[,]𝐵)(𝑦 < 𝑥 → ∃𝑧 ∈ 𝐿 𝑦 < 𝑧)))
13		simpr 110	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) ∧ (∀𝑦 ∈ 𝐿 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ (𝐴[,]𝐵)(𝑦 < 𝑥 → ∃𝑧 ∈ 𝐿 𝑦 < 𝑧))) ∧ 𝑞 ∈ 𝐿) → 𝑞 ∈ 𝐿)
14	3	ad3antrrr 492	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) ∧ (∀𝑦 ∈ 𝐿 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ (𝐴[,]𝐵)(𝑦 < 𝑥 → ∃𝑧 ∈ 𝐿 𝑦 < 𝑧))) ∧ 𝑞 ∈ 𝐿) → 𝐿 ⊆ (𝐴[,]𝐵))
15	14, 13	sseldd 3158	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) ∧ (∀𝑦 ∈ 𝐿 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ (𝐴[,]𝐵)(𝑦 < 𝑥 → ∃𝑧 ∈ 𝐿 𝑦 < 𝑧))) ∧ 𝑞 ∈ 𝐿) → 𝑞 ∈ (𝐴[,]𝐵))
16		rsp 2524	. . . . . . . . . . 11 ⊢ (∀𝑞 ∈ (𝐴[,]𝐵)(𝑞 ∈ 𝐿 ↔ ∃𝑟 ∈ 𝐿 𝑞 < 𝑟) → (𝑞 ∈ (𝐴[,]𝐵) → (𝑞 ∈ 𝐿 ↔ ∃𝑟 ∈ 𝐿 𝑞 < 𝑟)))
17	7, 16	syl 14	. . . . . . . . . 10 ⊢ (𝜑 → (𝑞 ∈ (𝐴[,]𝐵) → (𝑞 ∈ 𝐿 ↔ ∃𝑟 ∈ 𝐿 𝑞 < 𝑟)))
18	17	ad3antrrr 492	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) ∧ (∀𝑦 ∈ 𝐿 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ (𝐴[,]𝐵)(𝑦 < 𝑥 → ∃𝑧 ∈ 𝐿 𝑦 < 𝑧))) ∧ 𝑞 ∈ 𝐿) → (𝑞 ∈ (𝐴[,]𝐵) → (𝑞 ∈ 𝐿 ↔ ∃𝑟 ∈ 𝐿 𝑞 < 𝑟)))
19	15, 18	mpd 13	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) ∧ (∀𝑦 ∈ 𝐿 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ (𝐴[,]𝐵)(𝑦 < 𝑥 → ∃𝑧 ∈ 𝐿 𝑦 < 𝑧))) ∧ 𝑞 ∈ 𝐿) → (𝑞 ∈ 𝐿 ↔ ∃𝑟 ∈ 𝐿 𝑞 < 𝑟))
20	13, 19	mpbid 147	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) ∧ (∀𝑦 ∈ 𝐿 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ (𝐴[,]𝐵)(𝑦 < 𝑥 → ∃𝑧 ∈ 𝐿 𝑦 < 𝑧))) ∧ 𝑞 ∈ 𝐿) → ∃𝑟 ∈ 𝐿 𝑞 < 𝑟)
21		iccssre 9957	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴[,]𝐵) ⊆ ℝ)
22	1, 2, 21	syl2anc 411	. . . . . . . . . 10 ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ ℝ)
23	22	ad4antr 494	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) ∧ (∀𝑦 ∈ 𝐿 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ (𝐴[,]𝐵)(𝑦 < 𝑥 → ∃𝑧 ∈ 𝐿 𝑦 < 𝑧))) ∧ 𝑞 ∈ 𝐿) ∧ (𝑟 ∈ 𝐿 ∧ 𝑞 < 𝑟)) → (𝐴[,]𝐵) ⊆ ℝ)
24	15	adantr 276	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) ∧ (∀𝑦 ∈ 𝐿 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ (𝐴[,]𝐵)(𝑦 < 𝑥 → ∃𝑧 ∈ 𝐿 𝑦 < 𝑧))) ∧ 𝑞 ∈ 𝐿) ∧ (𝑟 ∈ 𝐿 ∧ 𝑞 < 𝑟)) → 𝑞 ∈ (𝐴[,]𝐵))
25	23, 24	sseldd 3158	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) ∧ (∀𝑦 ∈ 𝐿 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ (𝐴[,]𝐵)(𝑦 < 𝑥 → ∃𝑧 ∈ 𝐿 𝑦 < 𝑧))) ∧ 𝑞 ∈ 𝐿) ∧ (𝑟 ∈ 𝐿 ∧ 𝑞 < 𝑟)) → 𝑞 ∈ ℝ)
26	3	ad4antr 494	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) ∧ (∀𝑦 ∈ 𝐿 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ (𝐴[,]𝐵)(𝑦 < 𝑥 → ∃𝑧 ∈ 𝐿 𝑦 < 𝑧))) ∧ 𝑞 ∈ 𝐿) ∧ (𝑟 ∈ 𝐿 ∧ 𝑞 < 𝑟)) → 𝐿 ⊆ (𝐴[,]𝐵))
27		simprl 529	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) ∧ (∀𝑦 ∈ 𝐿 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ (𝐴[,]𝐵)(𝑦 < 𝑥 → ∃𝑧 ∈ 𝐿 𝑦 < 𝑧))) ∧ 𝑞 ∈ 𝐿) ∧ (𝑟 ∈ 𝐿 ∧ 𝑞 < 𝑟)) → 𝑟 ∈ 𝐿)
28	26, 27	sseldd 3158	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) ∧ (∀𝑦 ∈ 𝐿 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ (𝐴[,]𝐵)(𝑦 < 𝑥 → ∃𝑧 ∈ 𝐿 𝑦 < 𝑧))) ∧ 𝑞 ∈ 𝐿) ∧ (𝑟 ∈ 𝐿 ∧ 𝑞 < 𝑟)) → 𝑟 ∈ (𝐴[,]𝐵))
29	23, 28	sseldd 3158	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) ∧ (∀𝑦 ∈ 𝐿 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ (𝐴[,]𝐵)(𝑦 < 𝑥 → ∃𝑧 ∈ 𝐿 𝑦 < 𝑧))) ∧ 𝑞 ∈ 𝐿) ∧ (𝑟 ∈ 𝐿 ∧ 𝑞 < 𝑟)) → 𝑟 ∈ ℝ)
30		simp-4r 542	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) ∧ (∀𝑦 ∈ 𝐿 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ (𝐴[,]𝐵)(𝑦 < 𝑥 → ∃𝑧 ∈ 𝐿 𝑦 < 𝑧))) ∧ 𝑞 ∈ 𝐿) ∧ (𝑟 ∈ 𝐿 ∧ 𝑞 < 𝑟)) → 𝑥 ∈ (𝐴[,]𝐵))
31	23, 30	sseldd 3158	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) ∧ (∀𝑦 ∈ 𝐿 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ (𝐴[,]𝐵)(𝑦 < 𝑥 → ∃𝑧 ∈ 𝐿 𝑦 < 𝑧))) ∧ 𝑞 ∈ 𝐿) ∧ (𝑟 ∈ 𝐿 ∧ 𝑞 < 𝑟)) → 𝑥 ∈ ℝ)
32		simprr 531	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) ∧ (∀𝑦 ∈ 𝐿 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ (𝐴[,]𝐵)(𝑦 < 𝑥 → ∃𝑧 ∈ 𝐿 𝑦 < 𝑧))) ∧ 𝑞 ∈ 𝐿) ∧ (𝑟 ∈ 𝐿 ∧ 𝑞 < 𝑟)) → 𝑞 < 𝑟)
33		breq2 4009	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑟 → (𝑥 < 𝑦 ↔ 𝑥 < 𝑟))
34	33	notbid 667	. . . . . . . . . 10 ⊢ (𝑦 = 𝑟 → (¬ 𝑥 < 𝑦 ↔ ¬ 𝑥 < 𝑟))
35		simprl 529	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) ∧ (∀𝑦 ∈ 𝐿 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ (𝐴[,]𝐵)(𝑦 < 𝑥 → ∃𝑧 ∈ 𝐿 𝑦 < 𝑧))) → ∀𝑦 ∈ 𝐿 ¬ 𝑥 < 𝑦)
36	35	ad2antrr 488	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) ∧ (∀𝑦 ∈ 𝐿 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ (𝐴[,]𝐵)(𝑦 < 𝑥 → ∃𝑧 ∈ 𝐿 𝑦 < 𝑧))) ∧ 𝑞 ∈ 𝐿) ∧ (𝑟 ∈ 𝐿 ∧ 𝑞 < 𝑟)) → ∀𝑦 ∈ 𝐿 ¬ 𝑥 < 𝑦)
37	34, 36, 27	rspcdva 2848	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) ∧ (∀𝑦 ∈ 𝐿 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ (𝐴[,]𝐵)(𝑦 < 𝑥 → ∃𝑧 ∈ 𝐿 𝑦 < 𝑧))) ∧ 𝑞 ∈ 𝐿) ∧ (𝑟 ∈ 𝐿 ∧ 𝑞 < 𝑟)) → ¬ 𝑥 < 𝑟)
38	29, 31, 37	nltled 8080	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) ∧ (∀𝑦 ∈ 𝐿 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ (𝐴[,]𝐵)(𝑦 < 𝑥 → ∃𝑧 ∈ 𝐿 𝑦 < 𝑧))) ∧ 𝑞 ∈ 𝐿) ∧ (𝑟 ∈ 𝐿 ∧ 𝑞 < 𝑟)) → 𝑟 ≤ 𝑥)
39	25, 29, 31, 32, 38	ltletrd 8382	. . . . . . 7 ⊢ (((((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) ∧ (∀𝑦 ∈ 𝐿 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ (𝐴[,]𝐵)(𝑦 < 𝑥 → ∃𝑧 ∈ 𝐿 𝑦 < 𝑧))) ∧ 𝑞 ∈ 𝐿) ∧ (𝑟 ∈ 𝐿 ∧ 𝑞 < 𝑟)) → 𝑞 < 𝑥)
40	20, 39	rexlimddv 2599	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) ∧ (∀𝑦 ∈ 𝐿 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ (𝐴[,]𝐵)(𝑦 < 𝑥 → ∃𝑧 ∈ 𝐿 𝑦 < 𝑧))) ∧ 𝑞 ∈ 𝐿) → 𝑞 < 𝑥)
41	40	ralrimiva 2550	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) ∧ (∀𝑦 ∈ 𝐿 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ (𝐴[,]𝐵)(𝑦 < 𝑥 → ∃𝑧 ∈ 𝐿 𝑦 < 𝑧))) → ∀𝑞 ∈ 𝐿 𝑞 < 𝑥)
42		simpr 110	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) ∧ (∀𝑦 ∈ 𝐿 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ (𝐴[,]𝐵)(𝑦 < 𝑥 → ∃𝑧 ∈ 𝐿 𝑦 < 𝑧))) ∧ 𝑟 ∈ 𝑈) → 𝑟 ∈ 𝑈)
43		simplll 533	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) ∧ (∀𝑦 ∈ 𝐿 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ (𝐴[,]𝐵)(𝑦 < 𝑥 → ∃𝑧 ∈ 𝐿 𝑦 < 𝑧))) ∧ 𝑟 ∈ 𝑈) → 𝜑)
44	4	ad3antrrr 492	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) ∧ (∀𝑦 ∈ 𝐿 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ (𝐴[,]𝐵)(𝑦 < 𝑥 → ∃𝑧 ∈ 𝐿 𝑦 < 𝑧))) ∧ 𝑟 ∈ 𝑈) → 𝑈 ⊆ (𝐴[,]𝐵))
45	44, 42	sseldd 3158	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) ∧ (∀𝑦 ∈ 𝐿 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ (𝐴[,]𝐵)(𝑦 < 𝑥 → ∃𝑧 ∈ 𝐿 𝑦 < 𝑧))) ∧ 𝑟 ∈ 𝑈) → 𝑟 ∈ (𝐴[,]𝐵))
46		rsp 2524	. . . . . . . . . 10 ⊢ (∀𝑟 ∈ (𝐴[,]𝐵)(𝑟 ∈ 𝑈 ↔ ∃𝑞 ∈ 𝑈 𝑞 < 𝑟) → (𝑟 ∈ (𝐴[,]𝐵) → (𝑟 ∈ 𝑈 ↔ ∃𝑞 ∈ 𝑈 𝑞 < 𝑟)))
47	8, 46	syl 14	. . . . . . . . 9 ⊢ (𝜑 → (𝑟 ∈ (𝐴[,]𝐵) → (𝑟 ∈ 𝑈 ↔ ∃𝑞 ∈ 𝑈 𝑞 < 𝑟)))
48	43, 45, 47	sylc 62	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) ∧ (∀𝑦 ∈ 𝐿 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ (𝐴[,]𝐵)(𝑦 < 𝑥 → ∃𝑧 ∈ 𝐿 𝑦 < 𝑧))) ∧ 𝑟 ∈ 𝑈) → (𝑟 ∈ 𝑈 ↔ ∃𝑞 ∈ 𝑈 𝑞 < 𝑟))
49	42, 48	mpbid 147	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) ∧ (∀𝑦 ∈ 𝐿 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ (𝐴[,]𝐵)(𝑦 < 𝑥 → ∃𝑧 ∈ 𝐿 𝑦 < 𝑧))) ∧ 𝑟 ∈ 𝑈) → ∃𝑞 ∈ 𝑈 𝑞 < 𝑟)
50	22	ad4antr 494	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) ∧ (∀𝑦 ∈ 𝐿 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ (𝐴[,]𝐵)(𝑦 < 𝑥 → ∃𝑧 ∈ 𝐿 𝑦 < 𝑧))) ∧ 𝑟 ∈ 𝑈) ∧ (𝑞 ∈ 𝑈 ∧ 𝑞 < 𝑟)) → (𝐴[,]𝐵) ⊆ ℝ)
51		simp-4r 542	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) ∧ (∀𝑦 ∈ 𝐿 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ (𝐴[,]𝐵)(𝑦 < 𝑥 → ∃𝑧 ∈ 𝐿 𝑦 < 𝑧))) ∧ 𝑟 ∈ 𝑈) ∧ (𝑞 ∈ 𝑈 ∧ 𝑞 < 𝑟)) → 𝑥 ∈ (𝐴[,]𝐵))
52	50, 51	sseldd 3158	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) ∧ (∀𝑦 ∈ 𝐿 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ (𝐴[,]𝐵)(𝑦 < 𝑥 → ∃𝑧 ∈ 𝐿 𝑦 < 𝑧))) ∧ 𝑟 ∈ 𝑈) ∧ (𝑞 ∈ 𝑈 ∧ 𝑞 < 𝑟)) → 𝑥 ∈ ℝ)
53	4	ad4antr 494	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) ∧ (∀𝑦 ∈ 𝐿 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ (𝐴[,]𝐵)(𝑦 < 𝑥 → ∃𝑧 ∈ 𝐿 𝑦 < 𝑧))) ∧ 𝑟 ∈ 𝑈) ∧ (𝑞 ∈ 𝑈 ∧ 𝑞 < 𝑟)) → 𝑈 ⊆ (𝐴[,]𝐵))
54		simprl 529	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) ∧ (∀𝑦 ∈ 𝐿 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ (𝐴[,]𝐵)(𝑦 < 𝑥 → ∃𝑧 ∈ 𝐿 𝑦 < 𝑧))) ∧ 𝑟 ∈ 𝑈) ∧ (𝑞 ∈ 𝑈 ∧ 𝑞 < 𝑟)) → 𝑞 ∈ 𝑈)
55	53, 54	sseldd 3158	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) ∧ (∀𝑦 ∈ 𝐿 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ (𝐴[,]𝐵)(𝑦 < 𝑥 → ∃𝑧 ∈ 𝐿 𝑦 < 𝑧))) ∧ 𝑟 ∈ 𝑈) ∧ (𝑞 ∈ 𝑈 ∧ 𝑞 < 𝑟)) → 𝑞 ∈ (𝐴[,]𝐵))
56	50, 55	sseldd 3158	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) ∧ (∀𝑦 ∈ 𝐿 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ (𝐴[,]𝐵)(𝑦 < 𝑥 → ∃𝑧 ∈ 𝐿 𝑦 < 𝑧))) ∧ 𝑟 ∈ 𝑈) ∧ (𝑞 ∈ 𝑈 ∧ 𝑞 < 𝑟)) → 𝑞 ∈ ℝ)
57	45	adantr 276	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) ∧ (∀𝑦 ∈ 𝐿 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ (𝐴[,]𝐵)(𝑦 < 𝑥 → ∃𝑧 ∈ 𝐿 𝑦 < 𝑧))) ∧ 𝑟 ∈ 𝑈) ∧ (𝑞 ∈ 𝑈 ∧ 𝑞 < 𝑟)) → 𝑟 ∈ (𝐴[,]𝐵))
58	50, 57	sseldd 3158	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) ∧ (∀𝑦 ∈ 𝐿 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ (𝐴[,]𝐵)(𝑦 < 𝑥 → ∃𝑧 ∈ 𝐿 𝑦 < 𝑧))) ∧ 𝑟 ∈ 𝑈) ∧ (𝑞 ∈ 𝑈 ∧ 𝑞 < 𝑟)) → 𝑟 ∈ ℝ)
59	54	adantr 276	. . . . . . . . . 10 ⊢ ((((((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) ∧ (∀𝑦 ∈ 𝐿 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ (𝐴[,]𝐵)(𝑦 < 𝑥 → ∃𝑧 ∈ 𝐿 𝑦 < 𝑧))) ∧ 𝑟 ∈ 𝑈) ∧ (𝑞 ∈ 𝑈 ∧ 𝑞 < 𝑟)) ∧ 𝑞 < 𝑥) → 𝑞 ∈ 𝑈)
60	43	ad2antrr 488	. . . . . . . . . . 11 ⊢ ((((((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) ∧ (∀𝑦 ∈ 𝐿 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ (𝐴[,]𝐵)(𝑦 < 𝑥 → ∃𝑧 ∈ 𝐿 𝑦 < 𝑧))) ∧ 𝑟 ∈ 𝑈) ∧ (𝑞 ∈ 𝑈 ∧ 𝑞 < 𝑟)) ∧ 𝑞 < 𝑥) → 𝜑)
61		simpr 110	. . . . . . . . . . . . . 14 ⊢ ((((((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) ∧ (∀𝑦 ∈ 𝐿 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ (𝐴[,]𝐵)(𝑦 < 𝑥 → ∃𝑧 ∈ 𝐿 𝑦 < 𝑧))) ∧ 𝑟 ∈ 𝑈) ∧ (𝑞 ∈ 𝑈 ∧ 𝑞 < 𝑟)) ∧ 𝑞 < 𝑥) → 𝑞 < 𝑥)
62		breq1 4008	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = 𝑞 → (𝑦 < 𝑥 ↔ 𝑞 < 𝑥))
63		breq1 4008	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = 𝑞 → (𝑦 < 𝑧 ↔ 𝑞 < 𝑧))
64	63	rexbidv 2478	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = 𝑞 → (∃𝑧 ∈ 𝐿 𝑦 < 𝑧 ↔ ∃𝑧 ∈ 𝐿 𝑞 < 𝑧))
65	62, 64	imbi12d 234	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = 𝑞 → ((𝑦 < 𝑥 → ∃𝑧 ∈ 𝐿 𝑦 < 𝑧) ↔ (𝑞 < 𝑥 → ∃𝑧 ∈ 𝐿 𝑞 < 𝑧)))
66		simprr 531	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) ∧ (∀𝑦 ∈ 𝐿 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ (𝐴[,]𝐵)(𝑦 < 𝑥 → ∃𝑧 ∈ 𝐿 𝑦 < 𝑧))) → ∀𝑦 ∈ (𝐴[,]𝐵)(𝑦 < 𝑥 → ∃𝑧 ∈ 𝐿 𝑦 < 𝑧))
67	66	ad3antrrr 492	. . . . . . . . . . . . . . 15 ⊢ ((((((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) ∧ (∀𝑦 ∈ 𝐿 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ (𝐴[,]𝐵)(𝑦 < 𝑥 → ∃𝑧 ∈ 𝐿 𝑦 < 𝑧))) ∧ 𝑟 ∈ 𝑈) ∧ (𝑞 ∈ 𝑈 ∧ 𝑞 < 𝑟)) ∧ 𝑞 < 𝑥) → ∀𝑦 ∈ (𝐴[,]𝐵)(𝑦 < 𝑥 → ∃𝑧 ∈ 𝐿 𝑦 < 𝑧))
68	55	adantr 276	. . . . . . . . . . . . . . 15 ⊢ ((((((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) ∧ (∀𝑦 ∈ 𝐿 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ (𝐴[,]𝐵)(𝑦 < 𝑥 → ∃𝑧 ∈ 𝐿 𝑦 < 𝑧))) ∧ 𝑟 ∈ 𝑈) ∧ (𝑞 ∈ 𝑈 ∧ 𝑞 < 𝑟)) ∧ 𝑞 < 𝑥) → 𝑞 ∈ (𝐴[,]𝐵))
69	65, 67, 68	rspcdva 2848	. . . . . . . . . . . . . 14 ⊢ ((((((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) ∧ (∀𝑦 ∈ 𝐿 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ (𝐴[,]𝐵)(𝑦 < 𝑥 → ∃𝑧 ∈ 𝐿 𝑦 < 𝑧))) ∧ 𝑟 ∈ 𝑈) ∧ (𝑞 ∈ 𝑈 ∧ 𝑞 < 𝑟)) ∧ 𝑞 < 𝑥) → (𝑞 < 𝑥 → ∃𝑧 ∈ 𝐿 𝑞 < 𝑧))
70	61, 69	mpd 13	. . . . . . . . . . . . 13 ⊢ ((((((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) ∧ (∀𝑦 ∈ 𝐿 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ (𝐴[,]𝐵)(𝑦 < 𝑥 → ∃𝑧 ∈ 𝐿 𝑦 < 𝑧))) ∧ 𝑟 ∈ 𝑈) ∧ (𝑞 ∈ 𝑈 ∧ 𝑞 < 𝑟)) ∧ 𝑞 < 𝑥) → ∃𝑧 ∈ 𝐿 𝑞 < 𝑧)
71		breq2 4009	. . . . . . . . . . . . . 14 ⊢ (𝑧 = 𝑟 → (𝑞 < 𝑧 ↔ 𝑞 < 𝑟))
72	71	cbvrexv 2706	. . . . . . . . . . . . 13 ⊢ (∃𝑧 ∈ 𝐿 𝑞 < 𝑧 ↔ ∃𝑟 ∈ 𝐿 𝑞 < 𝑟)
73	70, 72	sylib 122	. . . . . . . . . . . 12 ⊢ ((((((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) ∧ (∀𝑦 ∈ 𝐿 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ (𝐴[,]𝐵)(𝑦 < 𝑥 → ∃𝑧 ∈ 𝐿 𝑦 < 𝑧))) ∧ 𝑟 ∈ 𝑈) ∧ (𝑞 ∈ 𝑈 ∧ 𝑞 < 𝑟)) ∧ 𝑞 < 𝑥) → ∃𝑟 ∈ 𝐿 𝑞 < 𝑟)
74	60, 68, 17	sylc 62	. . . . . . . . . . . 12 ⊢ ((((((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) ∧ (∀𝑦 ∈ 𝐿 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ (𝐴[,]𝐵)(𝑦 < 𝑥 → ∃𝑧 ∈ 𝐿 𝑦 < 𝑧))) ∧ 𝑟 ∈ 𝑈) ∧ (𝑞 ∈ 𝑈 ∧ 𝑞 < 𝑟)) ∧ 𝑞 < 𝑥) → (𝑞 ∈ 𝐿 ↔ ∃𝑟 ∈ 𝐿 𝑞 < 𝑟))
75	73, 74	mpbird 167	. . . . . . . . . . 11 ⊢ ((((((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) ∧ (∀𝑦 ∈ 𝐿 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ (𝐴[,]𝐵)(𝑦 < 𝑥 → ∃𝑧 ∈ 𝐿 𝑦 < 𝑧))) ∧ 𝑟 ∈ 𝑈) ∧ (𝑞 ∈ 𝑈 ∧ 𝑞 < 𝑟)) ∧ 𝑞 < 𝑥) → 𝑞 ∈ 𝐿)
76		disj 3473	. . . . . . . . . . . . 13 ⊢ ((𝐿 ∩ 𝑈) = ∅ ↔ ∀𝑞 ∈ 𝐿 ¬ 𝑞 ∈ 𝑈)
77	9, 76	sylib 122	. . . . . . . . . . . 12 ⊢ (𝜑 → ∀𝑞 ∈ 𝐿 ¬ 𝑞 ∈ 𝑈)
78	77	r19.21bi 2565	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐿) → ¬ 𝑞 ∈ 𝑈)
79	60, 75, 78	syl2anc 411	. . . . . . . . . 10 ⊢ ((((((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) ∧ (∀𝑦 ∈ 𝐿 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ (𝐴[,]𝐵)(𝑦 < 𝑥 → ∃𝑧 ∈ 𝐿 𝑦 < 𝑧))) ∧ 𝑟 ∈ 𝑈) ∧ (𝑞 ∈ 𝑈 ∧ 𝑞 < 𝑟)) ∧ 𝑞 < 𝑥) → ¬ 𝑞 ∈ 𝑈)
80	59, 79	pm2.65da 661	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) ∧ (∀𝑦 ∈ 𝐿 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ (𝐴[,]𝐵)(𝑦 < 𝑥 → ∃𝑧 ∈ 𝐿 𝑦 < 𝑧))) ∧ 𝑟 ∈ 𝑈) ∧ (𝑞 ∈ 𝑈 ∧ 𝑞 < 𝑟)) → ¬ 𝑞 < 𝑥)
81	52, 56, 80	nltled 8080	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) ∧ (∀𝑦 ∈ 𝐿 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ (𝐴[,]𝐵)(𝑦 < 𝑥 → ∃𝑧 ∈ 𝐿 𝑦 < 𝑧))) ∧ 𝑟 ∈ 𝑈) ∧ (𝑞 ∈ 𝑈 ∧ 𝑞 < 𝑟)) → 𝑥 ≤ 𝑞)
82		simprr 531	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) ∧ (∀𝑦 ∈ 𝐿 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ (𝐴[,]𝐵)(𝑦 < 𝑥 → ∃𝑧 ∈ 𝐿 𝑦 < 𝑧))) ∧ 𝑟 ∈ 𝑈) ∧ (𝑞 ∈ 𝑈 ∧ 𝑞 < 𝑟)) → 𝑞 < 𝑟)
83	52, 56, 58, 81, 82	lelttrd 8084	. . . . . . 7 ⊢ (((((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) ∧ (∀𝑦 ∈ 𝐿 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ (𝐴[,]𝐵)(𝑦 < 𝑥 → ∃𝑧 ∈ 𝐿 𝑦 < 𝑧))) ∧ 𝑟 ∈ 𝑈) ∧ (𝑞 ∈ 𝑈 ∧ 𝑞 < 𝑟)) → 𝑥 < 𝑟)
84	49, 83	rexlimddv 2599	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) ∧ (∀𝑦 ∈ 𝐿 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ (𝐴[,]𝐵)(𝑦 < 𝑥 → ∃𝑧 ∈ 𝐿 𝑦 < 𝑧))) ∧ 𝑟 ∈ 𝑈) → 𝑥 < 𝑟)
85	84	ralrimiva 2550	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) ∧ (∀𝑦 ∈ 𝐿 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ (𝐴[,]𝐵)(𝑦 < 𝑥 → ∃𝑧 ∈ 𝐿 𝑦 < 𝑧))) → ∀𝑟 ∈ 𝑈 𝑥 < 𝑟)
86	41, 85	jca 306	. . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) ∧ (∀𝑦 ∈ 𝐿 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ (𝐴[,]𝐵)(𝑦 < 𝑥 → ∃𝑧 ∈ 𝐿 𝑦 < 𝑧))) → (∀𝑞 ∈ 𝐿 𝑞 < 𝑥 ∧ ∀𝑟 ∈ 𝑈 𝑥 < 𝑟))
87	86	ex 115	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → ((∀𝑦 ∈ 𝐿 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ (𝐴[,]𝐵)(𝑦 < 𝑥 → ∃𝑧 ∈ 𝐿 𝑦 < 𝑧)) → (∀𝑞 ∈ 𝐿 𝑞 < 𝑥 ∧ ∀𝑟 ∈ 𝑈 𝑥 < 𝑟)))
88	87	reximdva 2579	. 2 ⊢ (𝜑 → (∃𝑥 ∈ (𝐴[,]𝐵)(∀𝑦 ∈ 𝐿 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ (𝐴[,]𝐵)(𝑦 < 𝑥 → ∃𝑧 ∈ 𝐿 𝑦 < 𝑧)) → ∃𝑥 ∈ (𝐴[,]𝐵)(∀𝑞 ∈ 𝐿 𝑞 < 𝑥 ∧ ∀𝑟 ∈ 𝑈 𝑥 < 𝑟)))
89	12, 88	mpd 13	1 ⊢ (𝜑 → ∃𝑥 ∈ (𝐴[,]𝐵)(∀𝑞 ∈ 𝐿 𝑞 < 𝑥 ∧ ∀𝑟 ∈ 𝑈 𝑥 < 𝑟))

Colors of variables: wff set class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 104 ↔ wb 105 ∨ wo 708 = wceq 1353 ∈ wcel 2148 ∀wral 2455 ∃wrex 2456 ∩ cin 3130 ⊆ wss 3131 ∅c0 3424 class class class wbr 4005 (class class class)co 5877 ℝcr 7812 < clt 7994 [,]cicc 9893

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-coll 4120 ax-sep 4123 ax-nul 4131 ax-pow 4176 ax-pr 4211 ax-un 4435 ax-setind 4538 ax-iinf 4589 ax-cnex 7904 ax-resscn 7905 ax-1cn 7906 ax-1re 7907 ax-icn 7908 ax-addcl 7909 ax-addrcl 7910 ax-mulcl 7911 ax-mulrcl 7912 ax-addcom 7913 ax-mulcom 7914 ax-addass 7915 ax-mulass 7916 ax-distr 7917 ax-i2m1 7918 ax-0lt1 7919 ax-1rid 7920 ax-0id 7921 ax-rnegex 7922 ax-precex 7923 ax-cnre 7924 ax-pre-ltirr 7925 ax-pre-ltwlin 7926 ax-pre-lttrn 7927 ax-pre-apti 7928 ax-pre-ltadd 7929 ax-pre-mulgt0 7930 ax-pre-mulext 7931 ax-arch 7932 ax-caucvg 7933 ax-pre-suploc 7934

This theorem depends on definitions: df-bi 117 df-dc 835 df-3or 979 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-reu 2462 df-rmo 2463 df-rab 2464 df-v 2741 df-sbc 2965 df-csb 3060 df-dif 3133 df-un 3135 df-in 3137 df-ss 3144 df-nul 3425 df-if 3537 df-pw 3579 df-sn 3600 df-pr 3601 df-op 3603 df-uni 3812 df-int 3847 df-iun 3890 df-br 4006 df-opab 4067 df-mpt 4068 df-tr 4104 df-id 4295 df-po 4298 df-iso 4299 df-iord 4368 df-on 4370 df-ilim 4371 df-suc 4373 df-iom 4592 df-xp 4634 df-rel 4635 df-cnv 4636 df-co 4637 df-dm 4638 df-rn 4639 df-res 4640 df-ima 4641 df-iota 5180 df-fun 5220 df-fn 5221 df-f 5222 df-f1 5223 df-fo 5224 df-f1o 5225 df-fv 5226 df-isom 5227 df-riota 5833 df-ov 5880 df-oprab 5881 df-mpo 5882 df-1st 6143 df-2nd 6144 df-recs 6308 df-frec 6394 df-sup 6985 df-inf 6986 df-pnf 7996 df-mnf 7997 df-xr 7998 df-ltxr 7999 df-le 8000 df-sub 8132 df-neg 8133 df-reap 8534 df-ap 8541 df-div 8632 df-inn 8922 df-2 8980 df-3 8981 df-4 8982 df-n0 9179 df-z 9256 df-uz 9531 df-rp 9656 df-icc 9897 df-seqfrec 10448 df-exp 10522 df-cj 10853 df-re 10854 df-im 10855 df-rsqrt 11009 df-abs 11010

This theorem is referenced by: dedekindicclemicc 14149

W3C validator