Theorem suplocexprlemru 7720

Description: Lemma for suplocexpr 7726. 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 7716	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐴 ⊆ P)
5		suplocexpr.b	. . . . . . . . . . . 12 ⊢ 𝐵 = ⟨∪ (1st “ 𝐴), {𝑢 ∈ Q ∣ ∃𝑤 ∈ ∩ (2nd “ 𝐴)𝑤 <Q 𝑢}⟩
6	5	suplocexprlem2b 7715	. . . . . . . . . . 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 4009	. . . . . . . . 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 2895	. . . . . . 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 7416	. . . . . . . 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 4009	. . . . . . . . . . . 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 2897	. . . . . . . . . 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 2895	. . . . . . . . 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 7366	. . . . . . . . . . . . . 14 ⊢ <Q ⊆ (Q × Q)
51	50	brel 4680	. . . . . . . . . . . . 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 7399	. . . . . . . . . . . 12 ⊢ <Q Or Q
57		sotr 4320	. . . . . . . . . . . 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 2897	. . . . 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 3131  ⟨cop 3597  ∪ cuni 3811  ∩ cint 3846   class class class wbr 4005   Or wor 4297   “ cima 4631  ‘cfv 5218  1^st c1st 6141  2^nd c2nd 6142  Qcnq 7281   <_Q cltq 7286  Pcnp 7292  <_P cltp 7296

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

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 2741  df-sbc 2965  df-csb 3060  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  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-eprel 4291  df-id 4295  df-po 4298  df-iso 4299  df-iord 4368  df-on 4370  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-ov 5880  df-oprab 5881  df-mpo 5882  df-1st 6143  df-2nd 6144  df-recs 6308  df-irdg 6373  df-1o 6419  df-oadd 6423  df-omul 6424  df-er 6537  df-ec 6539  df-qs 6543  df-ni 7305  df-pli 7306  df-mi 7307  df-lti 7308  df-plpq 7345  df-mpq 7346  df-enq 7348  df-nqqs 7349  df-plqqs 7350  df-mqqs 7351  df-1nqqs 7352  df-rq 7353  df-ltnqqs 7354  df-inp 7467  df-iltp 7471

This theorem is referenced by:  suplocexprlemex  7723