Theorem recexprlemloc 7911

Description: 𝐵 is located. Lemma for recexpr 7918. (Contributed by Jim Kingdon, 27-Dec-2019.)

Hypothesis

Ref	Expression
recexpr.1	⊢ 𝐵 = ⟨{𝑥 ∣ ∃𝑦(𝑥 <Q 𝑦 ∧ (*Q‘𝑦) ∈ (2nd ‘𝐴))}, {𝑥 ∣ ∃𝑦(𝑦 <Q 𝑥 ∧ (*Q‘𝑦) ∈ (1st ‘𝐴))}⟩

Assertion

Ref	Expression
recexprlemloc	⊢ (𝐴 ∈ P → ∀𝑞 ∈ Q ∀𝑟 ∈ Q (𝑞 <Q 𝑟 → (𝑞 ∈ (1st ‘𝐵) ∨ 𝑟 ∈ (2nd ‘𝐵))))

Distinct variable groups:   𝑟,𝑞,𝑥,𝑦,𝐴   𝐵,𝑞,𝑟,𝑥,𝑦

Proof of Theorem recexprlemloc

Dummy variables 𝑣 𝑢 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		prop 7755	. . . . . . . . 9 ⊢ (𝐴 ∈ P → ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P)
2		prnmaxl 7768	. . . . . . . . 9 ⊢ ((⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P ∧ (*Q‘𝑟) ∈ (1st ‘𝐴)) → ∃𝑢 ∈ (1st ‘𝐴)(*Q‘𝑟) <Q 𝑢)
3	1, 2	sylan 283	. . . . . . . 8 ⊢ ((𝐴 ∈ P ∧ (*Q‘𝑟) ∈ (1st ‘𝐴)) → ∃𝑢 ∈ (1st ‘𝐴)(*Q‘𝑟) <Q 𝑢)
4	3	adantlr 477	. . . . . . 7 ⊢ (((𝐴 ∈ P ∧ 𝑞 <Q 𝑟) ∧ (*Q‘𝑟) ∈ (1st ‘𝐴)) → ∃𝑢 ∈ (1st ‘𝐴)(*Q‘𝑟) <Q 𝑢)
5		simprr 533	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ P ∧ 𝑞 <Q 𝑟) ∧ (*Q‘𝑟) ∈ (1st ‘𝐴)) ∧ (𝑢 ∈ (1st ‘𝐴) ∧ (*Q‘𝑟) <Q 𝑢)) → (*Q‘𝑟) <Q 𝑢)
6		elprnql 7761	. . . . . . . . . . . . . 14 ⊢ ((⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P ∧ 𝑢 ∈ (1st ‘𝐴)) → 𝑢 ∈ Q)
7	1, 6	sylan 283	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ P ∧ 𝑢 ∈ (1st ‘𝐴)) → 𝑢 ∈ Q)
8	7	ad2ant2r 509	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ P ∧ 𝑞 <Q 𝑟) ∧ (𝑢 ∈ (1st ‘𝐴) ∧ (*Q‘𝑟) <Q 𝑢)) → 𝑢 ∈ Q)
9	8	adantlr 477	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ P ∧ 𝑞 <Q 𝑟) ∧ (*Q‘𝑟) ∈ (1st ‘𝐴)) ∧ (𝑢 ∈ (1st ‘𝐴) ∧ (*Q‘𝑟) <Q 𝑢)) → 𝑢 ∈ Q)
10		recrecnq 7674	. . . . . . . . . . 11 ⊢ (𝑢 ∈ Q → (*Q‘(*Q‘𝑢)) = 𝑢)
11	9, 10	syl 14	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ P ∧ 𝑞 <Q 𝑟) ∧ (*Q‘𝑟) ∈ (1st ‘𝐴)) ∧ (𝑢 ∈ (1st ‘𝐴) ∧ (*Q‘𝑟) <Q 𝑢)) → (*Q‘(*Q‘𝑢)) = 𝑢)
12	5, 11	breqtrrd 4121	. . . . . . . . 9 ⊢ ((((𝐴 ∈ P ∧ 𝑞 <Q 𝑟) ∧ (*Q‘𝑟) ∈ (1st ‘𝐴)) ∧ (𝑢 ∈ (1st ‘𝐴) ∧ (*Q‘𝑟) <Q 𝑢)) → (*Q‘𝑟) <Q (*Q‘(*Q‘𝑢)))
13		recclnq 7672	. . . . . . . . . . 11 ⊢ (𝑢 ∈ Q → (*Q‘𝑢) ∈ Q)
14	9, 13	syl 14	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ P ∧ 𝑞 <Q 𝑟) ∧ (*Q‘𝑟) ∈ (1st ‘𝐴)) ∧ (𝑢 ∈ (1st ‘𝐴) ∧ (*Q‘𝑟) <Q 𝑢)) → (*Q‘𝑢) ∈ Q)
15		ltrelnq 7645	. . . . . . . . . . . . . 14 ⊢ <Q ⊆ (Q × Q)
16	15	brel 4784	. . . . . . . . . . . . 13 ⊢ (𝑞 <Q 𝑟 → (𝑞 ∈ Q ∧ 𝑟 ∈ Q))
17	16	adantl 277	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ P ∧ 𝑞 <Q 𝑟) → (𝑞 ∈ Q ∧ 𝑟 ∈ Q))
18	17	ad2antrr 488	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ P ∧ 𝑞 <Q 𝑟) ∧ (*Q‘𝑟) ∈ (1st ‘𝐴)) ∧ (𝑢 ∈ (1st ‘𝐴) ∧ (*Q‘𝑟) <Q 𝑢)) → (𝑞 ∈ Q ∧ 𝑟 ∈ Q))
19	18	simprd 114	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ P ∧ 𝑞 <Q 𝑟) ∧ (*Q‘𝑟) ∈ (1st ‘𝐴)) ∧ (𝑢 ∈ (1st ‘𝐴) ∧ (*Q‘𝑟) <Q 𝑢)) → 𝑟 ∈ Q)
20		ltrnqg 7700	. . . . . . . . . 10 ⊢ (((*Q‘𝑢) ∈ Q ∧ 𝑟 ∈ Q) → ((*Q‘𝑢) <Q 𝑟 ↔ (*Q‘𝑟) <Q (*Q‘(*Q‘𝑢))))
21	14, 19, 20	syl2anc 411	. . . . . . . . 9 ⊢ ((((𝐴 ∈ P ∧ 𝑞 <Q 𝑟) ∧ (*Q‘𝑟) ∈ (1st ‘𝐴)) ∧ (𝑢 ∈ (1st ‘𝐴) ∧ (*Q‘𝑟) <Q 𝑢)) → ((*Q‘𝑢) <Q 𝑟 ↔ (*Q‘𝑟) <Q (*Q‘(*Q‘𝑢))))
22	12, 21	mpbird 167	. . . . . . . 8 ⊢ ((((𝐴 ∈ P ∧ 𝑞 <Q 𝑟) ∧ (*Q‘𝑟) ∈ (1st ‘𝐴)) ∧ (𝑢 ∈ (1st ‘𝐴) ∧ (*Q‘𝑟) <Q 𝑢)) → (*Q‘𝑢) <Q 𝑟)
23		simprl 531	. . . . . . . . 9 ⊢ ((((𝐴 ∈ P ∧ 𝑞 <Q 𝑟) ∧ (*Q‘𝑟) ∈ (1st ‘𝐴)) ∧ (𝑢 ∈ (1st ‘𝐴) ∧ (*Q‘𝑟) <Q 𝑢)) → 𝑢 ∈ (1st ‘𝐴))
24	11, 23	eqeltrd 2308	. . . . . . . 8 ⊢ ((((𝐴 ∈ P ∧ 𝑞 <Q 𝑟) ∧ (*Q‘𝑟) ∈ (1st ‘𝐴)) ∧ (𝑢 ∈ (1st ‘𝐴) ∧ (*Q‘𝑟) <Q 𝑢)) → (*Q‘(*Q‘𝑢)) ∈ (1st ‘𝐴))
25		breq1 4096	. . . . . . . . . . . 12 ⊢ (𝑦 = (*Q‘𝑢) → (𝑦 <Q 𝑟 ↔ (*Q‘𝑢) <Q 𝑟))
26		fveq2 5648	. . . . . . . . . . . . 13 ⊢ (𝑦 = (*Q‘𝑢) → (*Q‘𝑦) = (*Q‘(*Q‘𝑢)))
27	26	eleq1d 2300	. . . . . . . . . . . 12 ⊢ (𝑦 = (*Q‘𝑢) → ((*Q‘𝑦) ∈ (1st ‘𝐴) ↔ (*Q‘(*Q‘𝑢)) ∈ (1st ‘𝐴)))
28	25, 27	anbi12d 473	. . . . . . . . . . 11 ⊢ (𝑦 = (*Q‘𝑢) → ((𝑦 <Q 𝑟 ∧ (*Q‘𝑦) ∈ (1st ‘𝐴)) ↔ ((*Q‘𝑢) <Q 𝑟 ∧ (*Q‘(*Q‘𝑢)) ∈ (1st ‘𝐴))))
29	28	spcegv 2895	. . . . . . . . . 10 ⊢ ((*Q‘𝑢) ∈ Q → (((*Q‘𝑢) <Q 𝑟 ∧ (*Q‘(*Q‘𝑢)) ∈ (1st ‘𝐴)) → ∃𝑦(𝑦 <Q 𝑟 ∧ (*Q‘𝑦) ∈ (1st ‘𝐴))))
30		recexpr.1	. . . . . . . . . . 11 ⊢ 𝐵 = ⟨{𝑥 ∣ ∃𝑦(𝑥 <Q 𝑦 ∧ (*Q‘𝑦) ∈ (2nd ‘𝐴))}, {𝑥 ∣ ∃𝑦(𝑦 <Q 𝑥 ∧ (*Q‘𝑦) ∈ (1st ‘𝐴))}⟩
31	30	recexprlemelu 7903	. . . . . . . . . 10 ⊢ (𝑟 ∈ (2nd ‘𝐵) ↔ ∃𝑦(𝑦 <Q 𝑟 ∧ (*Q‘𝑦) ∈ (1st ‘𝐴)))
32	29, 31	imbitrrdi 162	. . . . . . . . 9 ⊢ ((*Q‘𝑢) ∈ Q → (((*Q‘𝑢) <Q 𝑟 ∧ (*Q‘(*Q‘𝑢)) ∈ (1st ‘𝐴)) → 𝑟 ∈ (2nd ‘𝐵)))
33	14, 32	syl 14	. . . . . . . 8 ⊢ ((((𝐴 ∈ P ∧ 𝑞 <Q 𝑟) ∧ (*Q‘𝑟) ∈ (1st ‘𝐴)) ∧ (𝑢 ∈ (1st ‘𝐴) ∧ (*Q‘𝑟) <Q 𝑢)) → (((*Q‘𝑢) <Q 𝑟 ∧ (*Q‘(*Q‘𝑢)) ∈ (1st ‘𝐴)) → 𝑟 ∈ (2nd ‘𝐵)))
34	22, 24, 33	mp2and 433	. . . . . . 7 ⊢ ((((𝐴 ∈ P ∧ 𝑞 <Q 𝑟) ∧ (*Q‘𝑟) ∈ (1st ‘𝐴)) ∧ (𝑢 ∈ (1st ‘𝐴) ∧ (*Q‘𝑟) <Q 𝑢)) → 𝑟 ∈ (2nd ‘𝐵))
35	4, 34	rexlimddv 2656	. . . . . 6 ⊢ (((𝐴 ∈ P ∧ 𝑞 <Q 𝑟) ∧ (*Q‘𝑟) ∈ (1st ‘𝐴)) → 𝑟 ∈ (2nd ‘𝐵))
36	35	olcd 742	. . . . 5 ⊢ (((𝐴 ∈ P ∧ 𝑞 <Q 𝑟) ∧ (*Q‘𝑟) ∈ (1st ‘𝐴)) → (𝑞 ∈ (1st ‘𝐵) ∨ 𝑟 ∈ (2nd ‘𝐵)))
37		prnminu 7769	. . . . . . . . 9 ⊢ ((⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P ∧ (*Q‘𝑞) ∈ (2nd ‘𝐴)) → ∃𝑣 ∈ (2nd ‘𝐴)𝑣 <Q (*Q‘𝑞))
38	1, 37	sylan 283	. . . . . . . 8 ⊢ ((𝐴 ∈ P ∧ (*Q‘𝑞) ∈ (2nd ‘𝐴)) → ∃𝑣 ∈ (2nd ‘𝐴)𝑣 <Q (*Q‘𝑞))
39	38	adantlr 477	. . . . . . 7 ⊢ (((𝐴 ∈ P ∧ 𝑞 <Q 𝑟) ∧ (*Q‘𝑞) ∈ (2nd ‘𝐴)) → ∃𝑣 ∈ (2nd ‘𝐴)𝑣 <Q (*Q‘𝑞))
40		elprnqu 7762	. . . . . . . . . . . . . 14 ⊢ ((⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P ∧ 𝑣 ∈ (2nd ‘𝐴)) → 𝑣 ∈ Q)
41	1, 40	sylan 283	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ P ∧ 𝑣 ∈ (2nd ‘𝐴)) → 𝑣 ∈ Q)
42	41	adantlr 477	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ P ∧ 𝑞 <Q 𝑟) ∧ 𝑣 ∈ (2nd ‘𝐴)) → 𝑣 ∈ Q)
43	42	ad2ant2r 509	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ P ∧ 𝑞 <Q 𝑟) ∧ (*Q‘𝑞) ∈ (2nd ‘𝐴)) ∧ (𝑣 ∈ (2nd ‘𝐴) ∧ 𝑣 <Q (*Q‘𝑞))) → 𝑣 ∈ Q)
44		recrecnq 7674	. . . . . . . . . . 11 ⊢ (𝑣 ∈ Q → (*Q‘(*Q‘𝑣)) = 𝑣)
45	43, 44	syl 14	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ P ∧ 𝑞 <Q 𝑟) ∧ (*Q‘𝑞) ∈ (2nd ‘𝐴)) ∧ (𝑣 ∈ (2nd ‘𝐴) ∧ 𝑣 <Q (*Q‘𝑞))) → (*Q‘(*Q‘𝑣)) = 𝑣)
46		simprr 533	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ P ∧ 𝑞 <Q 𝑟) ∧ (*Q‘𝑞) ∈ (2nd ‘𝐴)) ∧ (𝑣 ∈ (2nd ‘𝐴) ∧ 𝑣 <Q (*Q‘𝑞))) → 𝑣 <Q (*Q‘𝑞))
47	45, 46	eqbrtrd 4115	. . . . . . . . 9 ⊢ ((((𝐴 ∈ P ∧ 𝑞 <Q 𝑟) ∧ (*Q‘𝑞) ∈ (2nd ‘𝐴)) ∧ (𝑣 ∈ (2nd ‘𝐴) ∧ 𝑣 <Q (*Q‘𝑞))) → (*Q‘(*Q‘𝑣)) <Q (*Q‘𝑞))
48	17	ad2antrr 488	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ P ∧ 𝑞 <Q 𝑟) ∧ (*Q‘𝑞) ∈ (2nd ‘𝐴)) ∧ (𝑣 ∈ (2nd ‘𝐴) ∧ 𝑣 <Q (*Q‘𝑞))) → (𝑞 ∈ Q ∧ 𝑟 ∈ Q))
49	48	simpld 112	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ P ∧ 𝑞 <Q 𝑟) ∧ (*Q‘𝑞) ∈ (2nd ‘𝐴)) ∧ (𝑣 ∈ (2nd ‘𝐴) ∧ 𝑣 <Q (*Q‘𝑞))) → 𝑞 ∈ Q)
50		recclnq 7672	. . . . . . . . . . 11 ⊢ (𝑣 ∈ Q → (*Q‘𝑣) ∈ Q)
51	43, 50	syl 14	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ P ∧ 𝑞 <Q 𝑟) ∧ (*Q‘𝑞) ∈ (2nd ‘𝐴)) ∧ (𝑣 ∈ (2nd ‘𝐴) ∧ 𝑣 <Q (*Q‘𝑞))) → (*Q‘𝑣) ∈ Q)
52		ltrnqg 7700	. . . . . . . . . 10 ⊢ ((𝑞 ∈ Q ∧ (*Q‘𝑣) ∈ Q) → (𝑞 <Q (*Q‘𝑣) ↔ (*Q‘(*Q‘𝑣)) <Q (*Q‘𝑞)))
53	49, 51, 52	syl2anc 411	. . . . . . . . 9 ⊢ ((((𝐴 ∈ P ∧ 𝑞 <Q 𝑟) ∧ (*Q‘𝑞) ∈ (2nd ‘𝐴)) ∧ (𝑣 ∈ (2nd ‘𝐴) ∧ 𝑣 <Q (*Q‘𝑞))) → (𝑞 <Q (*Q‘𝑣) ↔ (*Q‘(*Q‘𝑣)) <Q (*Q‘𝑞)))
54	47, 53	mpbird 167	. . . . . . . 8 ⊢ ((((𝐴 ∈ P ∧ 𝑞 <Q 𝑟) ∧ (*Q‘𝑞) ∈ (2nd ‘𝐴)) ∧ (𝑣 ∈ (2nd ‘𝐴) ∧ 𝑣 <Q (*Q‘𝑞))) → 𝑞 <Q (*Q‘𝑣))
55		simprl 531	. . . . . . . . 9 ⊢ ((((𝐴 ∈ P ∧ 𝑞 <Q 𝑟) ∧ (*Q‘𝑞) ∈ (2nd ‘𝐴)) ∧ (𝑣 ∈ (2nd ‘𝐴) ∧ 𝑣 <Q (*Q‘𝑞))) → 𝑣 ∈ (2nd ‘𝐴))
56	45, 55	eqeltrd 2308	. . . . . . . 8 ⊢ ((((𝐴 ∈ P ∧ 𝑞 <Q 𝑟) ∧ (*Q‘𝑞) ∈ (2nd ‘𝐴)) ∧ (𝑣 ∈ (2nd ‘𝐴) ∧ 𝑣 <Q (*Q‘𝑞))) → (*Q‘(*Q‘𝑣)) ∈ (2nd ‘𝐴))
57		breq2 4097	. . . . . . . . . . . 12 ⊢ (𝑦 = (*Q‘𝑣) → (𝑞 <Q 𝑦 ↔ 𝑞 <Q (*Q‘𝑣)))
58		fveq2 5648	. . . . . . . . . . . . 13 ⊢ (𝑦 = (*Q‘𝑣) → (*Q‘𝑦) = (*Q‘(*Q‘𝑣)))
59	58	eleq1d 2300	. . . . . . . . . . . 12 ⊢ (𝑦 = (*Q‘𝑣) → ((*Q‘𝑦) ∈ (2nd ‘𝐴) ↔ (*Q‘(*Q‘𝑣)) ∈ (2nd ‘𝐴)))
60	57, 59	anbi12d 473	. . . . . . . . . . 11 ⊢ (𝑦 = (*Q‘𝑣) → ((𝑞 <Q 𝑦 ∧ (*Q‘𝑦) ∈ (2nd ‘𝐴)) ↔ (𝑞 <Q (*Q‘𝑣) ∧ (*Q‘(*Q‘𝑣)) ∈ (2nd ‘𝐴))))
61	60	spcegv 2895	. . . . . . . . . 10 ⊢ ((*Q‘𝑣) ∈ Q → ((𝑞 <Q (*Q‘𝑣) ∧ (*Q‘(*Q‘𝑣)) ∈ (2nd ‘𝐴)) → ∃𝑦(𝑞 <Q 𝑦 ∧ (*Q‘𝑦) ∈ (2nd ‘𝐴))))
62	30	recexprlemell 7902	. . . . . . . . . 10 ⊢ (𝑞 ∈ (1st ‘𝐵) ↔ ∃𝑦(𝑞 <Q 𝑦 ∧ (*Q‘𝑦) ∈ (2nd ‘𝐴)))
63	61, 62	imbitrrdi 162	. . . . . . . . 9 ⊢ ((*Q‘𝑣) ∈ Q → ((𝑞 <Q (*Q‘𝑣) ∧ (*Q‘(*Q‘𝑣)) ∈ (2nd ‘𝐴)) → 𝑞 ∈ (1st ‘𝐵)))
64	51, 63	syl 14	. . . . . . . 8 ⊢ ((((𝐴 ∈ P ∧ 𝑞 <Q 𝑟) ∧ (*Q‘𝑞) ∈ (2nd ‘𝐴)) ∧ (𝑣 ∈ (2nd ‘𝐴) ∧ 𝑣 <Q (*Q‘𝑞))) → ((𝑞 <Q (*Q‘𝑣) ∧ (*Q‘(*Q‘𝑣)) ∈ (2nd ‘𝐴)) → 𝑞 ∈ (1st ‘𝐵)))
65	54, 56, 64	mp2and 433	. . . . . . 7 ⊢ ((((𝐴 ∈ P ∧ 𝑞 <Q 𝑟) ∧ (*Q‘𝑞) ∈ (2nd ‘𝐴)) ∧ (𝑣 ∈ (2nd ‘𝐴) ∧ 𝑣 <Q (*Q‘𝑞))) → 𝑞 ∈ (1st ‘𝐵))
66	39, 65	rexlimddv 2656	. . . . . 6 ⊢ (((𝐴 ∈ P ∧ 𝑞 <Q 𝑟) ∧ (*Q‘𝑞) ∈ (2nd ‘𝐴)) → 𝑞 ∈ (1st ‘𝐵))
67	66	orcd 741	. . . . 5 ⊢ (((𝐴 ∈ P ∧ 𝑞 <Q 𝑟) ∧ (*Q‘𝑞) ∈ (2nd ‘𝐴)) → (𝑞 ∈ (1st ‘𝐵) ∨ 𝑟 ∈ (2nd ‘𝐵)))
68		ltrnqi 7701	. . . . . 6 ⊢ (𝑞 <Q 𝑟 → (*Q‘𝑟) <Q (*Q‘𝑞))
69		prloc 7771	. . . . . 6 ⊢ ((⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P ∧ (*Q‘𝑟) <Q (*Q‘𝑞)) → ((*Q‘𝑟) ∈ (1st ‘𝐴) ∨ (*Q‘𝑞) ∈ (2nd ‘𝐴)))
70	1, 68, 69	syl2an 289	. . . . 5 ⊢ ((𝐴 ∈ P ∧ 𝑞 <Q 𝑟) → ((*Q‘𝑟) ∈ (1st ‘𝐴) ∨ (*Q‘𝑞) ∈ (2nd ‘𝐴)))
71	36, 67, 70	mpjaodan 806	. . . 4 ⊢ ((𝐴 ∈ P ∧ 𝑞 <Q 𝑟) → (𝑞 ∈ (1st ‘𝐵) ∨ 𝑟 ∈ (2nd ‘𝐵)))
72	71	ex 115	. . 3 ⊢ (𝐴 ∈ P → (𝑞 <Q 𝑟 → (𝑞 ∈ (1st ‘𝐵) ∨ 𝑟 ∈ (2nd ‘𝐵))))
73	72	ralrimivw 2607	. 2 ⊢ (𝐴 ∈ P → ∀𝑟 ∈ Q (𝑞 <Q 𝑟 → (𝑞 ∈ (1st ‘𝐵) ∨ 𝑟 ∈ (2nd ‘𝐵))))
74	73	ralrimivw 2607	1 ⊢ (𝐴 ∈ P → ∀𝑞 ∈ Q ∀𝑟 ∈ Q (𝑞 <Q 𝑟 → (𝑞 ∈ (1st ‘𝐵) ∨ 𝑟 ∈ (2nd ‘𝐵))))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∨ wo 716   = wceq 1398  ∃wex 1541   ∈ wcel 2202  {cab 2217  ∀wral 2511  ∃wrex 2512  ⟨cop 3676   class class class wbr 4093  ‘cfv 5333  1^st c1st 6310  2^nd c2nd 6311  Qcnq 7560  *_Qcrq 7564   <_Q cltq 7565  Pcnp 7571

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4209  ax-sep 4212  ax-nul 4220  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-iinf 4692

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-ral 2516  df-rex 2517  df-reu 2518  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-tr 4193  df-eprel 4392  df-id 4396  df-iord 4469  df-on 4471  df-suc 4474  df-iom 4695  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-ov 6031  df-oprab 6032  df-mpo 6033  df-1st 6312  df-2nd 6313  df-recs 6514  df-irdg 6579  df-1o 6625  df-oadd 6629  df-omul 6630  df-er 6745  df-ec 6747  df-qs 6751  df-ni 7584  df-mi 7586  df-lti 7587  df-mpq 7625  df-enq 7627  df-nqqs 7628  df-mqqs 7630  df-1nqqs 7631  df-rq 7632  df-ltnqqs 7633  df-inp 7746

This theorem is referenced by:  recexprlempr  7912