Theorem suplocexprlemru 7717

Description: Lemma for suplocexpr 7723. The upper cut of the putative supremum is rounded. (Contributed by Jim Kingdon, 9-Jan-2024.)

Hypotheses

Ref	Expression
suplocexpr.m	⊢ (𝜑 → ∃𝑥 𝑥 ∈ 𝐴)
suplocexpr.ub	⊢ (𝜑 → ∃𝑥 ∈ P ∀𝑦 ∈ 𝐴 𝑦<P 𝑥)
suplocexpr.loc	⊢ (𝜑 → ∀𝑥 ∈ P ∀𝑦 ∈ P (𝑥<P 𝑦 → (∃𝑧 ∈ 𝐴 𝑥<P 𝑧 ∨ ∀𝑧 ∈ 𝐴 𝑧<P 𝑦)))
suplocexpr.b	⊢ 𝐵 = ⟨∪ (1st “ 𝐴), {𝑢 ∈ Q ∣ ∃𝑤 ∈ ∩ (2nd “ 𝐴)𝑤 <Q 𝑢}⟩

Assertion

Ref	Expression
suplocexprlemru	⊢ (𝜑 → ∀𝑟 ∈ Q (𝑟 ∈ (2nd ‘𝐵) ↔ ∃𝑞 ∈ Q (𝑞 <Q 𝑟 ∧ 𝑞 ∈ (2nd ‘𝐵))))

Distinct variable groups:   𝐴,𝑞,𝑢   𝑥,𝐴,𝑦   𝐵,𝑞,𝑤   𝜑,𝑞,𝑟,𝑤   𝜑,𝑥,𝑦   𝑢,𝑟,𝑤

Allowed substitution hints:   𝜑(𝑧,𝑢)   𝐴(𝑧,𝑤,𝑟)   𝐵(𝑥,𝑦,𝑧,𝑢,𝑟)

Proof of Theorem suplocexprlemru

Step	Hyp	Ref	Expression
1		suplocexpr.m	. . . . . . . . . . . 12 ⊢ (𝜑 → ∃𝑥 𝑥 ∈ 𝐴)
2		suplocexpr.ub	. . . . . . . . . . . 12 ⊢ (𝜑 → ∃𝑥 ∈ P ∀𝑦 ∈ 𝐴 𝑦<P 𝑥)
3		suplocexpr.loc	. . . . . . . . . . . 12 ⊢ (𝜑 → ∀𝑥 ∈ P ∀𝑦 ∈ P (𝑥<P 𝑦 → (∃𝑧 ∈ 𝐴 𝑥<P 𝑧 ∨ ∀𝑧 ∈ 𝐴 𝑧<P 𝑦)))
4	1, 2, 3	suplocexprlemss 7713	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐴 ⊆ P)
5		suplocexpr.b	. . . . . . . . . . . 12 ⊢ 𝐵 = ⟨∪ (1st “ 𝐴), {𝑢 ∈ Q ∣ ∃𝑤 ∈ ∩ (2nd “ 𝐴)𝑤 <Q 𝑢}⟩
6	5	suplocexprlem2b 7712	. . . . . . . . . . 11 ⊢ (𝐴 ⊆ P → (2nd ‘𝐵) = {𝑢 ∈ Q ∣ ∃𝑤 ∈ ∩ (2nd “ 𝐴)𝑤 <Q 𝑢})
7	4, 6	syl 14	. . . . . . . . . 10 ⊢ (𝜑 → (2nd ‘𝐵) = {𝑢 ∈ Q ∣ ∃𝑤 ∈ ∩ (2nd “ 𝐴)𝑤 <Q 𝑢})
8	7	eleq2d 2247	. . . . . . . . 9 ⊢ (𝜑 → (𝑟 ∈ (2nd ‘𝐵) ↔ 𝑟 ∈ {𝑢 ∈ Q ∣ ∃𝑤 ∈ ∩ (2nd “ 𝐴)𝑤 <Q 𝑢}))
9	8	adantr 276	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑟 ∈ Q) → (𝑟 ∈ (2nd ‘𝐵) ↔ 𝑟 ∈ {𝑢 ∈ Q ∣ ∃𝑤 ∈ ∩ (2nd “ 𝐴)𝑤 <Q 𝑢}))
10	9	biimpa 296	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑟 ∈ Q) ∧ 𝑟 ∈ (2nd ‘𝐵)) → 𝑟 ∈ {𝑢 ∈ Q ∣ ∃𝑤 ∈ ∩ (2nd “ 𝐴)𝑤 <Q 𝑢})
11		breq2 4007	. . . . . . . . 9 ⊢ (𝑢 = 𝑟 → (𝑤 <Q 𝑢 ↔ 𝑤 <Q 𝑟))
12	11	rexbidv 2478	. . . . . . . 8 ⊢ (𝑢 = 𝑟 → (∃𝑤 ∈ ∩ (2nd “ 𝐴)𝑤 <Q 𝑢 ↔ ∃𝑤 ∈ ∩ (2nd “ 𝐴)𝑤 <Q 𝑟))
13	12	elrab 2893	. . . . . . 7 ⊢ (𝑟 ∈ {𝑢 ∈ Q ∣ ∃𝑤 ∈ ∩ (2nd “ 𝐴)𝑤 <Q 𝑢} ↔ (𝑟 ∈ Q ∧ ∃𝑤 ∈ ∩ (2nd “ 𝐴)𝑤 <Q 𝑟))
14	10, 13	sylib 122	. . . . . 6 ⊢ (((𝜑 ∧ 𝑟 ∈ Q) ∧ 𝑟 ∈ (2nd ‘𝐵)) → (𝑟 ∈ Q ∧ ∃𝑤 ∈ ∩ (2nd “ 𝐴)𝑤 <Q 𝑟))
15	14	simprd 114	. . . . 5 ⊢ (((𝜑 ∧ 𝑟 ∈ Q) ∧ 𝑟 ∈ (2nd ‘𝐵)) → ∃𝑤 ∈ ∩ (2nd “ 𝐴)𝑤 <Q 𝑟)
16		ltbtwnnqq 7413	. . . . . . . 8 ⊢ (𝑤 <Q 𝑟 ↔ ∃𝑞 ∈ Q (𝑤 <Q 𝑞 ∧ 𝑞 <Q 𝑟))
17	16	biimpi 120	. . . . . . 7 ⊢ (𝑤 <Q 𝑟 → ∃𝑞 ∈ Q (𝑤 <Q 𝑞 ∧ 𝑞 <Q 𝑟))
18	17	ad2antll 491	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑟 ∈ Q) ∧ 𝑟 ∈ (2nd ‘𝐵)) ∧ (𝑤 ∈ ∩ (2nd “ 𝐴) ∧ 𝑤 <Q 𝑟)) → ∃𝑞 ∈ Q (𝑤 <Q 𝑞 ∧ 𝑞 <Q 𝑟))
19		simprr 531	. . . . . . . . 9 ⊢ ((((((𝜑 ∧ 𝑟 ∈ Q) ∧ 𝑟 ∈ (2nd ‘𝐵)) ∧ (𝑤 ∈ ∩ (2nd “ 𝐴) ∧ 𝑤 <Q 𝑟)) ∧ 𝑞 ∈ Q) ∧ (𝑤 <Q 𝑞 ∧ 𝑞 <Q 𝑟)) → 𝑞 <Q 𝑟)
20		breq2 4007	. . . . . . . . . . . 12 ⊢ (𝑢 = 𝑞 → (𝑤 <Q 𝑢 ↔ 𝑤 <Q 𝑞))
21	20	rexbidv 2478	. . . . . . . . . . 11 ⊢ (𝑢 = 𝑞 → (∃𝑤 ∈ ∩ (2nd “ 𝐴)𝑤 <Q 𝑢 ↔ ∃𝑤 ∈ ∩ (2nd “ 𝐴)𝑤 <Q 𝑞))
22		simplr 528	. . . . . . . . . . 11 ⊢ ((((((𝜑 ∧ 𝑟 ∈ Q) ∧ 𝑟 ∈ (2nd ‘𝐵)) ∧ (𝑤 ∈ ∩ (2nd “ 𝐴) ∧ 𝑤 <Q 𝑟)) ∧ 𝑞 ∈ Q) ∧ (𝑤 <Q 𝑞 ∧ 𝑞 <Q 𝑟)) → 𝑞 ∈ Q)
23		simprl 529	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑟 ∈ Q) ∧ 𝑟 ∈ (2nd ‘𝐵)) ∧ (𝑤 ∈ ∩ (2nd “ 𝐴) ∧ 𝑤 <Q 𝑟)) → 𝑤 ∈ ∩ (2nd “ 𝐴))
24	23	ad2antrr 488	. . . . . . . . . . . . 13 ⊢ ((((((𝜑 ∧ 𝑟 ∈ Q) ∧ 𝑟 ∈ (2nd ‘𝐵)) ∧ (𝑤 ∈ ∩ (2nd “ 𝐴) ∧ 𝑤 <Q 𝑟)) ∧ 𝑞 ∈ Q) ∧ (𝑤 <Q 𝑞 ∧ 𝑞 <Q 𝑟)) → 𝑤 ∈ ∩ (2nd “ 𝐴))
25		simprl 529	. . . . . . . . . . . . 13 ⊢ ((((((𝜑 ∧ 𝑟 ∈ Q) ∧ 𝑟 ∈ (2nd ‘𝐵)) ∧ (𝑤 ∈ ∩ (2nd “ 𝐴) ∧ 𝑤 <Q 𝑟)) ∧ 𝑞 ∈ Q) ∧ (𝑤 <Q 𝑞 ∧ 𝑞 <Q 𝑟)) → 𝑤 <Q 𝑞)
26	24, 25	jca 306	. . . . . . . . . . . 12 ⊢ ((((((𝜑 ∧ 𝑟 ∈ Q) ∧ 𝑟 ∈ (2nd ‘𝐵)) ∧ (𝑤 ∈ ∩ (2nd “ 𝐴) ∧ 𝑤 <Q 𝑟)) ∧ 𝑞 ∈ Q) ∧ (𝑤 <Q 𝑞 ∧ 𝑞 <Q 𝑟)) → (𝑤 ∈ ∩ (2nd “ 𝐴) ∧ 𝑤 <Q 𝑞))
27		rspe 2526	. . . . . . . . . . . 12 ⊢ ((𝑤 ∈ ∩ (2nd “ 𝐴) ∧ 𝑤 <Q 𝑞) → ∃𝑤 ∈ ∩ (2nd “ 𝐴)𝑤 <Q 𝑞)
28	26, 27	syl 14	. . . . . . . . . . 11 ⊢ ((((((𝜑 ∧ 𝑟 ∈ Q) ∧ 𝑟 ∈ (2nd ‘𝐵)) ∧ (𝑤 ∈ ∩ (2nd “ 𝐴) ∧ 𝑤 <Q 𝑟)) ∧ 𝑞 ∈ Q) ∧ (𝑤 <Q 𝑞 ∧ 𝑞 <Q 𝑟)) → ∃𝑤 ∈ ∩ (2nd “ 𝐴)𝑤 <Q 𝑞)
29	21, 22, 28	elrabd 2895	. . . . . . . . . 10 ⊢ ((((((𝜑 ∧ 𝑟 ∈ Q) ∧ 𝑟 ∈ (2nd ‘𝐵)) ∧ (𝑤 ∈ ∩ (2nd “ 𝐴) ∧ 𝑤 <Q 𝑟)) ∧ 𝑞 ∈ Q) ∧ (𝑤 <Q 𝑞 ∧ 𝑞 <Q 𝑟)) → 𝑞 ∈ {𝑢 ∈ Q ∣ ∃𝑤 ∈ ∩ (2nd “ 𝐴)𝑤 <Q 𝑢})
30	7	eleq2d 2247	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑞 ∈ (2nd ‘𝐵) ↔ 𝑞 ∈ {𝑢 ∈ Q ∣ ∃𝑤 ∈ ∩ (2nd “ 𝐴)𝑤 <Q 𝑢}))
31	30	ad5antr 496	. . . . . . . . . 10 ⊢ ((((((𝜑 ∧ 𝑟 ∈ Q) ∧ 𝑟 ∈ (2nd ‘𝐵)) ∧ (𝑤 ∈ ∩ (2nd “ 𝐴) ∧ 𝑤 <Q 𝑟)) ∧ 𝑞 ∈ Q) ∧ (𝑤 <Q 𝑞 ∧ 𝑞 <Q 𝑟)) → (𝑞 ∈ (2nd ‘𝐵) ↔ 𝑞 ∈ {𝑢 ∈ Q ∣ ∃𝑤 ∈ ∩ (2nd “ 𝐴)𝑤 <Q 𝑢}))
32	29, 31	mpbird 167	. . . . . . . . 9 ⊢ ((((((𝜑 ∧ 𝑟 ∈ Q) ∧ 𝑟 ∈ (2nd ‘𝐵)) ∧ (𝑤 ∈ ∩ (2nd “ 𝐴) ∧ 𝑤 <Q 𝑟)) ∧ 𝑞 ∈ Q) ∧ (𝑤 <Q 𝑞 ∧ 𝑞 <Q 𝑟)) → 𝑞 ∈ (2nd ‘𝐵))
33	19, 32	jca 306	. . . . . . . 8 ⊢ ((((((𝜑 ∧ 𝑟 ∈ Q) ∧ 𝑟 ∈ (2nd ‘𝐵)) ∧ (𝑤 ∈ ∩ (2nd “ 𝐴) ∧ 𝑤 <Q 𝑟)) ∧ 𝑞 ∈ Q) ∧ (𝑤 <Q 𝑞 ∧ 𝑞 <Q 𝑟)) → (𝑞 <Q 𝑟 ∧ 𝑞 ∈ (2nd ‘𝐵)))
34	33	ex 115	. . . . . . 7 ⊢ (((((𝜑 ∧ 𝑟 ∈ Q) ∧ 𝑟 ∈ (2nd ‘𝐵)) ∧ (𝑤 ∈ ∩ (2nd “ 𝐴) ∧ 𝑤 <Q 𝑟)) ∧ 𝑞 ∈ Q) → ((𝑤 <Q 𝑞 ∧ 𝑞 <Q 𝑟) → (𝑞 <Q 𝑟 ∧ 𝑞 ∈ (2nd ‘𝐵))))
35	34	reximdva 2579	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑟 ∈ Q) ∧ 𝑟 ∈ (2nd ‘𝐵)) ∧ (𝑤 ∈ ∩ (2nd “ 𝐴) ∧ 𝑤 <Q 𝑟)) → (∃𝑞 ∈ Q (𝑤 <Q 𝑞 ∧ 𝑞 <Q 𝑟) → ∃𝑞 ∈ Q (𝑞 <Q 𝑟 ∧ 𝑞 ∈ (2nd ‘𝐵))))
36	18, 35	mpd 13	. . . . 5 ⊢ ((((𝜑 ∧ 𝑟 ∈ Q) ∧ 𝑟 ∈ (2nd ‘𝐵)) ∧ (𝑤 ∈ ∩ (2nd “ 𝐴) ∧ 𝑤 <Q 𝑟)) → ∃𝑞 ∈ Q (𝑞 <Q 𝑟 ∧ 𝑞 ∈ (2nd ‘𝐵)))
37	15, 36	rexlimddv 2599	. . . 4 ⊢ (((𝜑 ∧ 𝑟 ∈ Q) ∧ 𝑟 ∈ (2nd ‘𝐵)) → ∃𝑞 ∈ Q (𝑞 <Q 𝑟 ∧ 𝑞 ∈ (2nd ‘𝐵)))
38	37	ex 115	. . 3 ⊢ ((𝜑 ∧ 𝑟 ∈ Q) → (𝑟 ∈ (2nd ‘𝐵) → ∃𝑞 ∈ Q (𝑞 <Q 𝑟 ∧ 𝑞 ∈ (2nd ‘𝐵))))
39		simpllr 534	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑟 ∈ Q) ∧ 𝑞 ∈ Q) ∧ (𝑞 <Q 𝑟 ∧ 𝑞 ∈ (2nd ‘𝐵))) → 𝑟 ∈ Q)
40		simprr 531	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑟 ∈ Q) ∧ 𝑞 ∈ Q) ∧ (𝑞 <Q 𝑟 ∧ 𝑞 ∈ (2nd ‘𝐵))) → 𝑞 ∈ (2nd ‘𝐵))
41	30	ad3antrrr 492	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑟 ∈ Q) ∧ 𝑞 ∈ Q) ∧ (𝑞 <Q 𝑟 ∧ 𝑞 ∈ (2nd ‘𝐵))) → (𝑞 ∈ (2nd ‘𝐵) ↔ 𝑞 ∈ {𝑢 ∈ Q ∣ ∃𝑤 ∈ ∩ (2nd “ 𝐴)𝑤 <Q 𝑢}))
42	40, 41	mpbid 147	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑟 ∈ Q) ∧ 𝑞 ∈ Q) ∧ (𝑞 <Q 𝑟 ∧ 𝑞 ∈ (2nd ‘𝐵))) → 𝑞 ∈ {𝑢 ∈ Q ∣ ∃𝑤 ∈ ∩ (2nd “ 𝐴)𝑤 <Q 𝑢})
43	21	elrab 2893	. . . . . . . . 9 ⊢ (𝑞 ∈ {𝑢 ∈ Q ∣ ∃𝑤 ∈ ∩ (2nd “ 𝐴)𝑤 <Q 𝑢} ↔ (𝑞 ∈ Q ∧ ∃𝑤 ∈ ∩ (2nd “ 𝐴)𝑤 <Q 𝑞))
44	42, 43	sylib 122	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑟 ∈ Q) ∧ 𝑞 ∈ Q) ∧ (𝑞 <Q 𝑟 ∧ 𝑞 ∈ (2nd ‘𝐵))) → (𝑞 ∈ Q ∧ ∃𝑤 ∈ ∩ (2nd “ 𝐴)𝑤 <Q 𝑞))
45	44	simprd 114	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑟 ∈ Q) ∧ 𝑞 ∈ Q) ∧ (𝑞 <Q 𝑟 ∧ 𝑞 ∈ (2nd ‘𝐵))) → ∃𝑤 ∈ ∩ (2nd “ 𝐴)𝑤 <Q 𝑞)
46		simpr 110	. . . . . . . . . . 11 ⊢ ((((((𝜑 ∧ 𝑟 ∈ Q) ∧ 𝑞 ∈ Q) ∧ (𝑞 <Q 𝑟 ∧ 𝑞 ∈ (2nd ‘𝐵))) ∧ 𝑤 ∈ ∩ (2nd “ 𝐴)) ∧ 𝑤 <Q 𝑞) → 𝑤 <Q 𝑞)
47		simprl 529	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑟 ∈ Q) ∧ 𝑞 ∈ Q) ∧ (𝑞 <Q 𝑟 ∧ 𝑞 ∈ (2nd ‘𝐵))) → 𝑞 <Q 𝑟)
48	47	ad2antrr 488	. . . . . . . . . . 11 ⊢ ((((((𝜑 ∧ 𝑟 ∈ Q) ∧ 𝑞 ∈ Q) ∧ (𝑞 <Q 𝑟 ∧ 𝑞 ∈ (2nd ‘𝐵))) ∧ 𝑤 ∈ ∩ (2nd “ 𝐴)) ∧ 𝑤 <Q 𝑞) → 𝑞 <Q 𝑟)
49	46, 48	jca 306	. . . . . . . . . 10 ⊢ ((((((𝜑 ∧ 𝑟 ∈ Q) ∧ 𝑞 ∈ Q) ∧ (𝑞 <Q 𝑟 ∧ 𝑞 ∈ (2nd ‘𝐵))) ∧ 𝑤 ∈ ∩ (2nd “ 𝐴)) ∧ 𝑤 <Q 𝑞) → (𝑤 <Q 𝑞 ∧ 𝑞 <Q 𝑟))
50		ltrelnq 7363	. . . . . . . . . . . . . 14 ⊢ <Q ⊆ (Q × Q)
51	50	brel 4678	. . . . . . . . . . . . 13 ⊢ (𝑤 <Q 𝑞 → (𝑤 ∈ Q ∧ 𝑞 ∈ Q))
52	51	simpld 112	. . . . . . . . . . . 12 ⊢ (𝑤 <Q 𝑞 → 𝑤 ∈ Q)
53	52	adantl 277	. . . . . . . . . . 11 ⊢ ((((((𝜑 ∧ 𝑟 ∈ Q) ∧ 𝑞 ∈ Q) ∧ (𝑞 <Q 𝑟 ∧ 𝑞 ∈ (2nd ‘𝐵))) ∧ 𝑤 ∈ ∩ (2nd “ 𝐴)) ∧ 𝑤 <Q 𝑞) → 𝑤 ∈ Q)
54		simp-4r 542	. . . . . . . . . . 11 ⊢ ((((((𝜑 ∧ 𝑟 ∈ Q) ∧ 𝑞 ∈ Q) ∧ (𝑞 <Q 𝑟 ∧ 𝑞 ∈ (2nd ‘𝐵))) ∧ 𝑤 ∈ ∩ (2nd “ 𝐴)) ∧ 𝑤 <Q 𝑞) → 𝑞 ∈ Q)
55	39	ad2antrr 488	. . . . . . . . . . 11 ⊢ ((((((𝜑 ∧ 𝑟 ∈ Q) ∧ 𝑞 ∈ Q) ∧ (𝑞 <Q 𝑟 ∧ 𝑞 ∈ (2nd ‘𝐵))) ∧ 𝑤 ∈ ∩ (2nd “ 𝐴)) ∧ 𝑤 <Q 𝑞) → 𝑟 ∈ Q)
56		ltsonq 7396	. . . . . . . . . . . 12 ⊢ <Q Or Q
57		sotr 4318	. . . . . . . . . . . 12 ⊢ (( <Q Or Q ∧ (𝑤 ∈ Q ∧ 𝑞 ∈ Q ∧ 𝑟 ∈ Q)) → ((𝑤 <Q 𝑞 ∧ 𝑞 <Q 𝑟) → 𝑤 <Q 𝑟))
58	56, 57	mpan 424	. . . . . . . . . . 11 ⊢ ((𝑤 ∈ Q ∧ 𝑞 ∈ Q ∧ 𝑟 ∈ Q) → ((𝑤 <Q 𝑞 ∧ 𝑞 <Q 𝑟) → 𝑤 <Q 𝑟))
59	53, 54, 55, 58	syl3anc 1238	. . . . . . . . . 10 ⊢ ((((((𝜑 ∧ 𝑟 ∈ Q) ∧ 𝑞 ∈ Q) ∧ (𝑞 <Q 𝑟 ∧ 𝑞 ∈ (2nd ‘𝐵))) ∧ 𝑤 ∈ ∩ (2nd “ 𝐴)) ∧ 𝑤 <Q 𝑞) → ((𝑤 <Q 𝑞 ∧ 𝑞 <Q 𝑟) → 𝑤 <Q 𝑟))
60	49, 59	mpd 13	. . . . . . . . 9 ⊢ ((((((𝜑 ∧ 𝑟 ∈ Q) ∧ 𝑞 ∈ Q) ∧ (𝑞 <Q 𝑟 ∧ 𝑞 ∈ (2nd ‘𝐵))) ∧ 𝑤 ∈ ∩ (2nd “ 𝐴)) ∧ 𝑤 <Q 𝑞) → 𝑤 <Q 𝑟)
61	60	ex 115	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑟 ∈ Q) ∧ 𝑞 ∈ Q) ∧ (𝑞 <Q 𝑟 ∧ 𝑞 ∈ (2nd ‘𝐵))) ∧ 𝑤 ∈ ∩ (2nd “ 𝐴)) → (𝑤 <Q 𝑞 → 𝑤 <Q 𝑟))
62	61	reximdva 2579	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑟 ∈ Q) ∧ 𝑞 ∈ Q) ∧ (𝑞 <Q 𝑟 ∧ 𝑞 ∈ (2nd ‘𝐵))) → (∃𝑤 ∈ ∩ (2nd “ 𝐴)𝑤 <Q 𝑞 → ∃𝑤 ∈ ∩ (2nd “ 𝐴)𝑤 <Q 𝑟))
63	45, 62	mpd 13	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑟 ∈ Q) ∧ 𝑞 ∈ Q) ∧ (𝑞 <Q 𝑟 ∧ 𝑞 ∈ (2nd ‘𝐵))) → ∃𝑤 ∈ ∩ (2nd “ 𝐴)𝑤 <Q 𝑟)
64	12, 39, 63	elrabd 2895	. . . . 5 ⊢ ((((𝜑 ∧ 𝑟 ∈ Q) ∧ 𝑞 ∈ Q) ∧ (𝑞 <Q 𝑟 ∧ 𝑞 ∈ (2nd ‘𝐵))) → 𝑟 ∈ {𝑢 ∈ Q ∣ ∃𝑤 ∈ ∩ (2nd “ 𝐴)𝑤 <Q 𝑢})
65	8	ad3antrrr 492	. . . . 5 ⊢ ((((𝜑 ∧ 𝑟 ∈ Q) ∧ 𝑞 ∈ Q) ∧ (𝑞 <Q 𝑟 ∧ 𝑞 ∈ (2nd ‘𝐵))) → (𝑟 ∈ (2nd ‘𝐵) ↔ 𝑟 ∈ {𝑢 ∈ Q ∣ ∃𝑤 ∈ ∩ (2nd “ 𝐴)𝑤 <Q 𝑢}))
66	64, 65	mpbird 167	. . . 4 ⊢ ((((𝜑 ∧ 𝑟 ∈ Q) ∧ 𝑞 ∈ Q) ∧ (𝑞 <Q 𝑟 ∧ 𝑞 ∈ (2nd ‘𝐵))) → 𝑟 ∈ (2nd ‘𝐵))
67	66	rexlimdva2 2597	. . 3 ⊢ ((𝜑 ∧ 𝑟 ∈ Q) → (∃𝑞 ∈ Q (𝑞 <Q 𝑟 ∧ 𝑞 ∈ (2nd ‘𝐵)) → 𝑟 ∈ (2nd ‘𝐵)))
68	38, 67	impbid 129	. 2 ⊢ ((𝜑 ∧ 𝑟 ∈ Q) → (𝑟 ∈ (2nd ‘𝐵) ↔ ∃𝑞 ∈ Q (𝑞 <Q 𝑟 ∧ 𝑞 ∈ (2nd ‘𝐵))))
69	68	ralrimiva 2550	1 ⊢ (𝜑 → ∀𝑟 ∈ Q (𝑟 ∈ (2nd ‘𝐵) ↔ ∃𝑞 ∈ Q (𝑞 <Q 𝑟 ∧ 𝑞 ∈ (2nd ‘𝐵))))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∨ wo 708   ∧ w3a 978   = wceq 1353  ∃wex 1492   ∈ wcel 2148  ∀wral 2455  ∃wrex 2456  {crab 2459   ⊆ wss 3129  ⟨cop 3595  ∪ cuni 3809  ∩ cint 3844   class class class wbr 4003   Or wor 4295   “ cima 4629  ‘cfv 5216  1^st c1st 6138  2^nd c2nd 6139  Qcnq 7278   <_Q cltq 7283  Pcnp 7289  <_P cltp 7293

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 4118  ax-sep 4121  ax-nul 4129  ax-pow 4174  ax-pr 4209  ax-un 4433  ax-setind 4536  ax-iinf 4587

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-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-int 3845  df-iun 3888  df-br 4004  df-opab 4065  df-mpt 4066  df-tr 4102  df-eprel 4289  df-id 4293  df-po 4296  df-iso 4297  df-iord 4366  df-on 4368  df-suc 4371  df-iom 4590  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-rn 4637  df-res 4638  df-ima 4639  df-iota 5178  df-fun 5218  df-fn 5219  df-f 5220  df-f1 5221  df-fo 5222  df-f1o 5223  df-fv 5224  df-ov 5877  df-oprab 5878  df-mpo 5879  df-1st 6140  df-2nd 6141  df-recs 6305  df-irdg 6370  df-1o 6416  df-oadd 6420  df-omul 6421  df-er 6534  df-ec 6536  df-qs 6540  df-ni 7302  df-pli 7303  df-mi 7304  df-lti 7305  df-plpq 7342  df-mpq 7343  df-enq 7345  df-nqqs 7346  df-plqqs 7347  df-mqqs 7348  df-1nqqs 7349  df-rq 7350  df-ltnqqs 7351  df-inp 7464  df-iltp 7468

This theorem is referenced by:  suplocexprlemex  7720