Theorem suplocexprlemru 7681

Description: Lemma for suplocexpr 7687. 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 7677	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐴 ⊆ P)
5		suplocexpr.b	. . . . . . . . . . . 12 ⊢ 𝐵 = ⟨∪ (1st “ 𝐴), {𝑢 ∈ Q ∣ ∃𝑤 ∈ ∩ (2nd “ 𝐴)𝑤 <Q 𝑢}⟩
6	5	suplocexprlem2b 7676	. . . . . . . . . . 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 2240	. . . . . . . . 9 ⊢ (𝜑 → (𝑟 ∈ (2nd ‘𝐵) ↔ 𝑟 ∈ {𝑢 ∈ Q ∣ ∃𝑤 ∈ ∩ (2nd “ 𝐴)𝑤 <Q 𝑢}))
9	8	adantr 274	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑟 ∈ Q) → (𝑟 ∈ (2nd ‘𝐵) ↔ 𝑟 ∈ {𝑢 ∈ Q ∣ ∃𝑤 ∈ ∩ (2nd “ 𝐴)𝑤 <Q 𝑢}))
10	9	biimpa 294	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑟 ∈ Q) ∧ 𝑟 ∈ (2nd ‘𝐵)) → 𝑟 ∈ {𝑢 ∈ Q ∣ ∃𝑤 ∈ ∩ (2nd “ 𝐴)𝑤 <Q 𝑢})
11		breq2 3993	. . . . . . . . 9 ⊢ (𝑢 = 𝑟 → (𝑤 <Q 𝑢 ↔ 𝑤 <Q 𝑟))
12	11	rexbidv 2471	. . . . . . . 8 ⊢ (𝑢 = 𝑟 → (∃𝑤 ∈ ∩ (2nd “ 𝐴)𝑤 <Q 𝑢 ↔ ∃𝑤 ∈ ∩ (2nd “ 𝐴)𝑤 <Q 𝑟))
13	12	elrab 2886	. . . . . . 7 ⊢ (𝑟 ∈ {𝑢 ∈ Q ∣ ∃𝑤 ∈ ∩ (2nd “ 𝐴)𝑤 <Q 𝑢} ↔ (𝑟 ∈ Q ∧ ∃𝑤 ∈ ∩ (2nd “ 𝐴)𝑤 <Q 𝑟))
14	10, 13	sylib 121	. . . . . 6 ⊢ (((𝜑 ∧ 𝑟 ∈ Q) ∧ 𝑟 ∈ (2nd ‘𝐵)) → (𝑟 ∈ Q ∧ ∃𝑤 ∈ ∩ (2nd “ 𝐴)𝑤 <Q 𝑟))
15	14	simprd 113	. . . . 5 ⊢ (((𝜑 ∧ 𝑟 ∈ Q) ∧ 𝑟 ∈ (2nd ‘𝐵)) → ∃𝑤 ∈ ∩ (2nd “ 𝐴)𝑤 <Q 𝑟)
16		ltbtwnnqq 7377	. . . . . . . 8 ⊢ (𝑤 <Q 𝑟 ↔ ∃𝑞 ∈ Q (𝑤 <Q 𝑞 ∧ 𝑞 <Q 𝑟))
17	16	biimpi 119	. . . . . . 7 ⊢ (𝑤 <Q 𝑟 → ∃𝑞 ∈ Q (𝑤 <Q 𝑞 ∧ 𝑞 <Q 𝑟))
18	17	ad2antll 488	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑟 ∈ Q) ∧ 𝑟 ∈ (2nd ‘𝐵)) ∧ (𝑤 ∈ ∩ (2nd “ 𝐴) ∧ 𝑤 <Q 𝑟)) → ∃𝑞 ∈ Q (𝑤 <Q 𝑞 ∧ 𝑞 <Q 𝑟))
19		simprr 527	. . . . . . . . 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 3993	. . . . . . . . . . . 12 ⊢ (𝑢 = 𝑞 → (𝑤 <Q 𝑢 ↔ 𝑤 <Q 𝑞))
21	20	rexbidv 2471	. . . . . . . . . . 11 ⊢ (𝑢 = 𝑞 → (∃𝑤 ∈ ∩ (2nd “ 𝐴)𝑤 <Q 𝑢 ↔ ∃𝑤 ∈ ∩ (2nd “ 𝐴)𝑤 <Q 𝑞))
22		simplr 525	. . . . . . . . . . 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 526	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑟 ∈ Q) ∧ 𝑟 ∈ (2nd ‘𝐵)) ∧ (𝑤 ∈ ∩ (2nd “ 𝐴) ∧ 𝑤 <Q 𝑟)) → 𝑤 ∈ ∩ (2nd “ 𝐴))
24	23	ad2antrr 485	. . . . . . . . . . . . 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 526	. . . . . . . . . . . . 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 304	. . . . . . . . . . . 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 2519	. . . . . . . . . . . 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 2888	. . . . . . . . . 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 2240	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑞 ∈ (2nd ‘𝐵) ↔ 𝑞 ∈ {𝑢 ∈ Q ∣ ∃𝑤 ∈ ∩ (2nd “ 𝐴)𝑤 <Q 𝑢}))
31	30	ad5antr 493	. . . . . . . . . 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 166	. . . . . . . . 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 304	. . . . . . . 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 114	. . . . . . 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 2572	. . . . . 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 2592	. . . 4 ⊢ (((𝜑 ∧ 𝑟 ∈ Q) ∧ 𝑟 ∈ (2nd ‘𝐵)) → ∃𝑞 ∈ Q (𝑞 <Q 𝑟 ∧ 𝑞 ∈ (2nd ‘𝐵)))
38	37	ex 114	. . 3 ⊢ ((𝜑 ∧ 𝑟 ∈ Q) → (𝑟 ∈ (2nd ‘𝐵) → ∃𝑞 ∈ Q (𝑞 <Q 𝑟 ∧ 𝑞 ∈ (2nd ‘𝐵))))
39		simpllr 529	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑟 ∈ Q) ∧ 𝑞 ∈ Q) ∧ (𝑞 <Q 𝑟 ∧ 𝑞 ∈ (2nd ‘𝐵))) → 𝑟 ∈ Q)
40		simprr 527	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑟 ∈ Q) ∧ 𝑞 ∈ Q) ∧ (𝑞 <Q 𝑟 ∧ 𝑞 ∈ (2nd ‘𝐵))) → 𝑞 ∈ (2nd ‘𝐵))
41	30	ad3antrrr 489	. . . . . . . . . 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 146	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑟 ∈ Q) ∧ 𝑞 ∈ Q) ∧ (𝑞 <Q 𝑟 ∧ 𝑞 ∈ (2nd ‘𝐵))) → 𝑞 ∈ {𝑢 ∈ Q ∣ ∃𝑤 ∈ ∩ (2nd “ 𝐴)𝑤 <Q 𝑢})
43	21	elrab 2886	. . . . . . . . 9 ⊢ (𝑞 ∈ {𝑢 ∈ Q ∣ ∃𝑤 ∈ ∩ (2nd “ 𝐴)𝑤 <Q 𝑢} ↔ (𝑞 ∈ Q ∧ ∃𝑤 ∈ ∩ (2nd “ 𝐴)𝑤 <Q 𝑞))
44	42, 43	sylib 121	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑟 ∈ Q) ∧ 𝑞 ∈ Q) ∧ (𝑞 <Q 𝑟 ∧ 𝑞 ∈ (2nd ‘𝐵))) → (𝑞 ∈ Q ∧ ∃𝑤 ∈ ∩ (2nd “ 𝐴)𝑤 <Q 𝑞))
45	44	simprd 113	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑟 ∈ Q) ∧ 𝑞 ∈ Q) ∧ (𝑞 <Q 𝑟 ∧ 𝑞 ∈ (2nd ‘𝐵))) → ∃𝑤 ∈ ∩ (2nd “ 𝐴)𝑤 <Q 𝑞)
46		simpr 109	. . . . . . . . . . 11 ⊢ ((((((𝜑 ∧ 𝑟 ∈ Q) ∧ 𝑞 ∈ Q) ∧ (𝑞 <Q 𝑟 ∧ 𝑞 ∈ (2nd ‘𝐵))) ∧ 𝑤 ∈ ∩ (2nd “ 𝐴)) ∧ 𝑤 <Q 𝑞) → 𝑤 <Q 𝑞)
47		simprl 526	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑟 ∈ Q) ∧ 𝑞 ∈ Q) ∧ (𝑞 <Q 𝑟 ∧ 𝑞 ∈ (2nd ‘𝐵))) → 𝑞 <Q 𝑟)
48	47	ad2antrr 485	. . . . . . . . . . 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 304	. . . . . . . . . 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 7327	. . . . . . . . . . . . . 14 ⊢ <Q ⊆ (Q × Q)
51	50	brel 4663	. . . . . . . . . . . . 13 ⊢ (𝑤 <Q 𝑞 → (𝑤 ∈ Q ∧ 𝑞 ∈ Q))
52	51	simpld 111	. . . . . . . . . . . 12 ⊢ (𝑤 <Q 𝑞 → 𝑤 ∈ Q)
53	52	adantl 275	. . . . . . . . . . 11 ⊢ ((((((𝜑 ∧ 𝑟 ∈ Q) ∧ 𝑞 ∈ Q) ∧ (𝑞 <Q 𝑟 ∧ 𝑞 ∈ (2nd ‘𝐵))) ∧ 𝑤 ∈ ∩ (2nd “ 𝐴)) ∧ 𝑤 <Q 𝑞) → 𝑤 ∈ Q)
54		simp-4r 537	. . . . . . . . . . 11 ⊢ ((((((𝜑 ∧ 𝑟 ∈ Q) ∧ 𝑞 ∈ Q) ∧ (𝑞 <Q 𝑟 ∧ 𝑞 ∈ (2nd ‘𝐵))) ∧ 𝑤 ∈ ∩ (2nd “ 𝐴)) ∧ 𝑤 <Q 𝑞) → 𝑞 ∈ Q)
55	39	ad2antrr 485	. . . . . . . . . . 11 ⊢ ((((((𝜑 ∧ 𝑟 ∈ Q) ∧ 𝑞 ∈ Q) ∧ (𝑞 <Q 𝑟 ∧ 𝑞 ∈ (2nd ‘𝐵))) ∧ 𝑤 ∈ ∩ (2nd “ 𝐴)) ∧ 𝑤 <Q 𝑞) → 𝑟 ∈ Q)
56		ltsonq 7360	. . . . . . . . . . . 12 ⊢ <Q Or Q
57		sotr 4303	. . . . . . . . . . . 12 ⊢ (( <Q Or Q ∧ (𝑤 ∈ Q ∧ 𝑞 ∈ Q ∧ 𝑟 ∈ Q)) → ((𝑤 <Q 𝑞 ∧ 𝑞 <Q 𝑟) → 𝑤 <Q 𝑟))
58	56, 57	mpan 422	. . . . . . . . . . 11 ⊢ ((𝑤 ∈ Q ∧ 𝑞 ∈ Q ∧ 𝑟 ∈ Q) → ((𝑤 <Q 𝑞 ∧ 𝑞 <Q 𝑟) → 𝑤 <Q 𝑟))
59	53, 54, 55, 58	syl3anc 1233	. . . . . . . . . 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 114	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑟 ∈ Q) ∧ 𝑞 ∈ Q) ∧ (𝑞 <Q 𝑟 ∧ 𝑞 ∈ (2nd ‘𝐵))) ∧ 𝑤 ∈ ∩ (2nd “ 𝐴)) → (𝑤 <Q 𝑞 → 𝑤 <Q 𝑟))
62	61	reximdva 2572	. . . . . . 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 2888	. . . . 5 ⊢ ((((𝜑 ∧ 𝑟 ∈ Q) ∧ 𝑞 ∈ Q) ∧ (𝑞 <Q 𝑟 ∧ 𝑞 ∈ (2nd ‘𝐵))) → 𝑟 ∈ {𝑢 ∈ Q ∣ ∃𝑤 ∈ ∩ (2nd “ 𝐴)𝑤 <Q 𝑢})
65	8	ad3antrrr 489	. . . . 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 166	. . . 4 ⊢ ((((𝜑 ∧ 𝑟 ∈ Q) ∧ 𝑞 ∈ Q) ∧ (𝑞 <Q 𝑟 ∧ 𝑞 ∈ (2nd ‘𝐵))) → 𝑟 ∈ (2nd ‘𝐵))
67	66	rexlimdva2 2590	. . 3 ⊢ ((𝜑 ∧ 𝑟 ∈ Q) → (∃𝑞 ∈ Q (𝑞 <Q 𝑟 ∧ 𝑞 ∈ (2nd ‘𝐵)) → 𝑟 ∈ (2nd ‘𝐵)))
68	38, 67	impbid 128	. 2 ⊢ ((𝜑 ∧ 𝑟 ∈ Q) → (𝑟 ∈ (2nd ‘𝐵) ↔ ∃𝑞 ∈ Q (𝑞 <Q 𝑟 ∧ 𝑞 ∈ (2nd ‘𝐵))))
69	68	ralrimiva 2543	1 ⊢ (𝜑 → ∀𝑟 ∈ Q (𝑟 ∈ (2nd ‘𝐵) ↔ ∃𝑞 ∈ Q (𝑞 <Q 𝑟 ∧ 𝑞 ∈ (2nd ‘𝐵))))

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   ∨ wo 703   ∧ w3a 973   = wceq 1348  ∃wex 1485   ∈ wcel 2141  ∀wral 2448  ∃wrex 2449  {crab 2452   ⊆ wss 3121  ⟨cop 3586  ∪ cuni 3796  ∩ cint 3831   class class class wbr 3989   Or wor 4280   “ cima 4614  ‘cfv 5198  1^st c1st 6117  2^nd c2nd 6118  Qcnq 7242   <_Q cltq 7247  Pcnp 7253  <_P cltp 7257

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4104  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-iinf 4572

This theorem depends on definitions:  df-bi 116  df-dc 830  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-tr 4088  df-eprel 4274  df-id 4278  df-po 4281  df-iso 4282  df-iord 4351  df-on 4353  df-suc 4356  df-iom 4575  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-ov 5856  df-oprab 5857  df-mpo 5858  df-1st 6119  df-2nd 6120  df-recs 6284  df-irdg 6349  df-1o 6395  df-oadd 6399  df-omul 6400  df-er 6513  df-ec 6515  df-qs 6519  df-ni 7266  df-pli 7267  df-mi 7268  df-lti 7269  df-plpq 7306  df-mpq 7307  df-enq 7309  df-nqqs 7310  df-plqqs 7311  df-mqqs 7312  df-1nqqs 7313  df-rq 7314  df-ltnqqs 7315  df-inp 7428  df-iltp 7432

This theorem is referenced by:  suplocexprlemex  7684