Theorem suplocexprlemru 7902

Description: Lemma for suplocexpr 7908. 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 7898	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐴 ⊆ P)
5		suplocexpr.b	. . . . . . . . . . . 12 ⊢ 𝐵 = ⟨∪ (1st “ 𝐴), {𝑢 ∈ Q ∣ ∃𝑤 ∈ ∩ (2nd “ 𝐴)𝑤 <Q 𝑢}⟩
6	5	suplocexprlem2b 7897	. . . . . . . . . . 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 2299	. . . . . . . . 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 4086	. . . . . . . . 9 ⊢ (𝑢 = 𝑟 → (𝑤 <Q 𝑢 ↔ 𝑤 <Q 𝑟))
12	11	rexbidv 2531	. . . . . . . 8 ⊢ (𝑢 = 𝑟 → (∃𝑤 ∈ ∩ (2nd “ 𝐴)𝑤 <Q 𝑢 ↔ ∃𝑤 ∈ ∩ (2nd “ 𝐴)𝑤 <Q 𝑟))
13	12	elrab 2959	. . . . . . 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 7598	. . . . . . . 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 4086	. . . . . . . . . . . 12 ⊢ (𝑢 = 𝑞 → (𝑤 <Q 𝑢 ↔ 𝑤 <Q 𝑞))
21	20	rexbidv 2531	. . . . . . . . . . 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 2579	. . . . . . . . . . . 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 2961	. . . . . . . . . 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 2299	. . . . . . . . . . 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 2632	. . . . . 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 2653	. . . 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 2959	. . . . . . . . 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 7548	. . . . . . . . . . . . . 14 ⊢ <Q ⊆ (Q × Q)
51	50	brel 4770	. . . . . . . . . . . . 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 7581	. . . . . . . . . . . 12 ⊢ <Q Or Q
57		sotr 4408	. . . . . . . . . . . 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 1271	. . . . . . . . . 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 2632	. . . . . . 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 2961	. . . . 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 2651	. . 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 2603	1 ⊢ (𝜑 → ∀𝑟 ∈ Q (𝑟 ∈ (2nd ‘𝐵) ↔ ∃𝑞 ∈ Q (𝑞 <Q 𝑟 ∧ 𝑞 ∈ (2nd ‘𝐵))))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∨ wo 713   ∧ w3a 1002   = wceq 1395  ∃wex 1538   ∈ wcel 2200  ∀wral 2508  ∃wrex 2509  {crab 2512   ⊆ wss 3197  ⟨cop 3669  ∪ cuni 3887  ∩ cint 3922   class class class wbr 4082   Or wor 4385   “ cima 4721  ‘cfv 5317  1^st c1st 6282  2^nd c2nd 6283  Qcnq 7463   <_Q cltq 7468  Pcnp 7474  <_P cltp 7478

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4198  ax-sep 4201  ax-nul 4209  ax-pow 4257  ax-pr 4292  ax-un 4523  ax-setind 4628  ax-iinf 4679

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-int 3923  df-iun 3966  df-br 4083  df-opab 4145  df-mpt 4146  df-tr 4182  df-eprel 4379  df-id 4383  df-po 4386  df-iso 4387  df-iord 4456  df-on 4458  df-suc 4461  df-iom 4682  df-xp 4724  df-rel 4725  df-cnv 4726  df-co 4727  df-dm 4728  df-rn 4729  df-res 4730  df-ima 4731  df-iota 5277  df-fun 5319  df-fn 5320  df-f 5321  df-f1 5322  df-fo 5323  df-f1o 5324  df-fv 5325  df-ov 6003  df-oprab 6004  df-mpo 6005  df-1st 6284  df-2nd 6285  df-recs 6449  df-irdg 6514  df-1o 6560  df-oadd 6564  df-omul 6565  df-er 6678  df-ec 6680  df-qs 6684  df-ni 7487  df-pli 7488  df-mi 7489  df-lti 7490  df-plpq 7527  df-mpq 7528  df-enq 7530  df-nqqs 7531  df-plqqs 7532  df-mqqs 7533  df-1nqqs 7534  df-rq 7535  df-ltnqqs 7536  df-inp 7649  df-iltp 7653

This theorem is referenced by:  suplocexprlemex  7905