Theorem suplocexprlemru 7527

Description: Lemma for suplocexpr 7533. 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 7523	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐴 ⊆ P)
5		suplocexpr.b	. . . . . . . . . . . 12 ⊢ 𝐵 = ⟨∪ (1st “ 𝐴), {𝑢 ∈ Q ∣ ∃𝑤 ∈ ∩ (2nd “ 𝐴)𝑤 <Q 𝑢}⟩
6	5	suplocexprlem2b 7522	. . . . . . . . . . 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 2209	. . . . . . . . 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 3933	. . . . . . . . 9 ⊢ (𝑢 = 𝑟 → (𝑤 <Q 𝑢 ↔ 𝑤 <Q 𝑟))
12	11	rexbidv 2438	. . . . . . . 8 ⊢ (𝑢 = 𝑟 → (∃𝑤 ∈ ∩ (2nd “ 𝐴)𝑤 <Q 𝑢 ↔ ∃𝑤 ∈ ∩ (2nd “ 𝐴)𝑤 <Q 𝑟))
13	12	elrab 2840	. . . . . . 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 7223	. . . . . . . 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 482	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑟 ∈ Q) ∧ 𝑟 ∈ (2nd ‘𝐵)) ∧ (𝑤 ∈ ∩ (2nd “ 𝐴) ∧ 𝑤 <Q 𝑟)) → ∃𝑞 ∈ Q (𝑤 <Q 𝑞 ∧ 𝑞 <Q 𝑟))
19		simprr 521	. . . . . . . . 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 3933	. . . . . . . . . . . 12 ⊢ (𝑢 = 𝑞 → (𝑤 <Q 𝑢 ↔ 𝑤 <Q 𝑞))
21	20	rexbidv 2438	. . . . . . . . . . 11 ⊢ (𝑢 = 𝑞 → (∃𝑤 ∈ ∩ (2nd “ 𝐴)𝑤 <Q 𝑢 ↔ ∃𝑤 ∈ ∩ (2nd “ 𝐴)𝑤 <Q 𝑞))
22		simplr 519	. . . . . . . . . . 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 520	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑟 ∈ Q) ∧ 𝑟 ∈ (2nd ‘𝐵)) ∧ (𝑤 ∈ ∩ (2nd “ 𝐴) ∧ 𝑤 <Q 𝑟)) → 𝑤 ∈ ∩ (2nd “ 𝐴))
24	23	ad2antrr 479	. . . . . . . . . . . . 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 520	. . . . . . . . . . . . 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 2481	. . . . . . . . . . . 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 2842	. . . . . . . . . 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 2209	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑞 ∈ (2nd ‘𝐵) ↔ 𝑞 ∈ {𝑢 ∈ Q ∣ ∃𝑤 ∈ ∩ (2nd “ 𝐴)𝑤 <Q 𝑢}))
31	30	ad5antr 487	. . . . . . . . . 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 2534	. . . . . 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 2554	. . . 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 523	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑟 ∈ Q) ∧ 𝑞 ∈ Q) ∧ (𝑞 <Q 𝑟 ∧ 𝑞 ∈ (2nd ‘𝐵))) → 𝑟 ∈ Q)
40		simprr 521	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑟 ∈ Q) ∧ 𝑞 ∈ Q) ∧ (𝑞 <Q 𝑟 ∧ 𝑞 ∈ (2nd ‘𝐵))) → 𝑞 ∈ (2nd ‘𝐵))
41	30	ad3antrrr 483	. . . . . . . . . 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 2840	. . . . . . . . 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 520	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑟 ∈ Q) ∧ 𝑞 ∈ Q) ∧ (𝑞 <Q 𝑟 ∧ 𝑞 ∈ (2nd ‘𝐵))) → 𝑞 <Q 𝑟)
48	47	ad2antrr 479	. . . . . . . . . . 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 7173	. . . . . . . . . . . . . 14 ⊢ <Q ⊆ (Q × Q)
51	50	brel 4591	. . . . . . . . . . . . 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 531	. . . . . . . . . . 11 ⊢ ((((((𝜑 ∧ 𝑟 ∈ Q) ∧ 𝑞 ∈ Q) ∧ (𝑞 <Q 𝑟 ∧ 𝑞 ∈ (2nd ‘𝐵))) ∧ 𝑤 ∈ ∩ (2nd “ 𝐴)) ∧ 𝑤 <Q 𝑞) → 𝑞 ∈ Q)
55	39	ad2antrr 479	. . . . . . . . . . 11 ⊢ ((((((𝜑 ∧ 𝑟 ∈ Q) ∧ 𝑞 ∈ Q) ∧ (𝑞 <Q 𝑟 ∧ 𝑞 ∈ (2nd ‘𝐵))) ∧ 𝑤 ∈ ∩ (2nd “ 𝐴)) ∧ 𝑤 <Q 𝑞) → 𝑟 ∈ Q)
56		ltsonq 7206	. . . . . . . . . . . 12 ⊢ <Q Or Q
57		sotr 4240	. . . . . . . . . . . 12 ⊢ (( <Q Or Q ∧ (𝑤 ∈ Q ∧ 𝑞 ∈ Q ∧ 𝑟 ∈ Q)) → ((𝑤 <Q 𝑞 ∧ 𝑞 <Q 𝑟) → 𝑤 <Q 𝑟))
58	56, 57	mpan 420	. . . . . . . . . . 11 ⊢ ((𝑤 ∈ Q ∧ 𝑞 ∈ Q ∧ 𝑟 ∈ Q) → ((𝑤 <Q 𝑞 ∧ 𝑞 <Q 𝑟) → 𝑤 <Q 𝑟))
59	53, 54, 55, 58	syl3anc 1216	. . . . . . . . . 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 2534	. . . . . . 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 2842	. . . . 5 ⊢ ((((𝜑 ∧ 𝑟 ∈ Q) ∧ 𝑞 ∈ Q) ∧ (𝑞 <Q 𝑟 ∧ 𝑞 ∈ (2nd ‘𝐵))) → 𝑟 ∈ {𝑢 ∈ Q ∣ ∃𝑤 ∈ ∩ (2nd “ 𝐴)𝑤 <Q 𝑢})
65	8	ad3antrrr 483	. . . . 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 2552	. . 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 2505	1 ⊢ (𝜑 → ∀𝑟 ∈ Q (𝑟 ∈ (2nd ‘𝐵) ↔ ∃𝑞 ∈ Q (𝑞 <Q 𝑟 ∧ 𝑞 ∈ (2nd ‘𝐵))))

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   ∨ wo 697   ∧ w3a 962   = wceq 1331  ∃wex 1468   ∈ wcel 1480  ∀wral 2416  ∃wrex 2417  {crab 2420   ⊆ wss 3071  ⟨cop 3530  ∪ cuni 3736  ∩ cint 3771   class class class wbr 3929   Or wor 4217   “ cima 4542  ‘cfv 5123  1^st c1st 6036  2^nd c2nd 6037  Qcnq 7088   <_Q cltq 7093  Pcnp 7099  <_P cltp 7103

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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4043  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-iinf 4502

This theorem depends on definitions:  df-bi 116  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-ral 2421  df-rex 2422  df-reu 2423  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-tr 4027  df-eprel 4211  df-id 4215  df-po 4218  df-iso 4219  df-iord 4288  df-on 4290  df-suc 4293  df-iom 4505  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-ov 5777  df-oprab 5778  df-mpo 5779  df-1st 6038  df-2nd 6039  df-recs 6202  df-irdg 6267  df-1o 6313  df-oadd 6317  df-omul 6318  df-er 6429  df-ec 6431  df-qs 6435  df-ni 7112  df-pli 7113  df-mi 7114  df-lti 7115  df-plpq 7152  df-mpq 7153  df-enq 7155  df-nqqs 7156  df-plqqs 7157  df-mqqs 7158  df-1nqqs 7159  df-rq 7160  df-ltnqqs 7161  df-inp 7274  df-iltp 7278

This theorem is referenced by:  suplocexprlemex  7530