Theorem recexprlemloc 7988

Description: 𝐵 is located. Lemma for recexpr 7995. (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 7832	. . . . . . . . 9 ⊢ (𝐴 ∈ P → ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P)
2		prnmaxl 7845	. . . . . . . . 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 481	. . . . . . 7 ⊢ (((𝐴 ∈ P ∧ 𝑞 <Q 𝑟) ∧ (*Q‘𝑟) ∈ (1st ‘𝐴)) → ∃𝑢 ∈ (1st ‘𝐴)(*Q‘𝑟) <Q 𝑢)
5		simprr 537	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ P ∧ 𝑞 <Q 𝑟) ∧ (*Q‘𝑟) ∈ (1st ‘𝐴)) ∧ (𝑢 ∈ (1st ‘𝐴) ∧ (*Q‘𝑟) <Q 𝑢)) → (*Q‘𝑟) <Q 𝑢)
6		elprnql 7838	. . . . . . . . . . . . . 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 513	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ P ∧ 𝑞 <Q 𝑟) ∧ (𝑢 ∈ (1st ‘𝐴) ∧ (*Q‘𝑟) <Q 𝑢)) → 𝑢 ∈ Q)
9	8	adantlr 481	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ P ∧ 𝑞 <Q 𝑟) ∧ (*Q‘𝑟) ∈ (1st ‘𝐴)) ∧ (𝑢 ∈ (1st ‘𝐴) ∧ (*Q‘𝑟) <Q 𝑢)) → 𝑢 ∈ Q)
10		recrecnq 7751	. . . . . . . . . . 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 4153	. . . . . . . . 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 7749	. . . . . . . . . . 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 7722	. . . . . . . . . . . . . 14 ⊢ <Q ⊆ (Q × Q)
16	15	brel 4822	. . . . . . . . . . . . 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 492	. . . . . . . . . . 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 7777	. . . . . . . . . 10 ⊢ (((*Q‘𝑢) ∈ Q ∧ 𝑟 ∈ Q) → ((*Q‘𝑢) <Q 𝑟 ↔ (*Q‘𝑟) <Q (*Q‘(*Q‘𝑢))))
21	14, 19, 20	syl2anc 415	. . . . . . . . 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 535	. . . . . . . . 9 ⊢ ((((𝐴 ∈ P ∧ 𝑞 <Q 𝑟) ∧ (*Q‘𝑟) ∈ (1st ‘𝐴)) ∧ (𝑢 ∈ (1st ‘𝐴) ∧ (*Q‘𝑟) <Q 𝑢)) → 𝑢 ∈ (1st ‘𝐴))
24	11, 23	eqeltrd 2315	. . . . . . . 8 ⊢ ((((𝐴 ∈ P ∧ 𝑞 <Q 𝑟) ∧ (*Q‘𝑟) ∈ (1st ‘𝐴)) ∧ (𝑢 ∈ (1st ‘𝐴) ∧ (*Q‘𝑟) <Q 𝑢)) → (*Q‘(*Q‘𝑢)) ∈ (1st ‘𝐴))
25		breq1 4128	. . . . . . . . . . . 12 ⊢ (𝑦 = (*Q‘𝑢) → (𝑦 <Q 𝑟 ↔ (*Q‘𝑢) <Q 𝑟))
26		fveq2 5690	. . . . . . . . . . . . 13 ⊢ (𝑦 = (*Q‘𝑢) → (*Q‘𝑦) = (*Q‘(*Q‘𝑢)))
27	26	eleq1d 2307	. . . . . . . . . . . 12 ⊢ (𝑦 = (*Q‘𝑢) → ((*Q‘𝑦) ∈ (1st ‘𝐴) ↔ (*Q‘(*Q‘𝑢)) ∈ (1st ‘𝐴)))
28	25, 27	anbi12d 477	. . . . . . . . . . 11 ⊢ (𝑦 = (*Q‘𝑢) → ((𝑦 <Q 𝑟 ∧ (*Q‘𝑦) ∈ (1st ‘𝐴)) ↔ ((*Q‘𝑢) <Q 𝑟 ∧ (*Q‘(*Q‘𝑢)) ∈ (1st ‘𝐴))))
29	28	spcegv 2913	. . . . . . . . . 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 7980	. . . . . . . . . 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 437	. . . . . . 7 ⊢ ((((𝐴 ∈ P ∧ 𝑞 <Q 𝑟) ∧ (*Q‘𝑟) ∈ (1st ‘𝐴)) ∧ (𝑢 ∈ (1st ‘𝐴) ∧ (*Q‘𝑟) <Q 𝑢)) → 𝑟 ∈ (2nd ‘𝐵))
35	4, 34	rexlimddv 2673	. . . . . 6 ⊢ (((𝐴 ∈ P ∧ 𝑞 <Q 𝑟) ∧ (*Q‘𝑟) ∈ (1st ‘𝐴)) → 𝑟 ∈ (2nd ‘𝐵))
36	35	olcd 746	. . . . 5 ⊢ (((𝐴 ∈ P ∧ 𝑞 <Q 𝑟) ∧ (*Q‘𝑟) ∈ (1st ‘𝐴)) → (𝑞 ∈ (1st ‘𝐵) ∨ 𝑟 ∈ (2nd ‘𝐵)))
37		prnminu 7846	. . . . . . . . 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 481	. . . . . . 7 ⊢ (((𝐴 ∈ P ∧ 𝑞 <Q 𝑟) ∧ (*Q‘𝑞) ∈ (2nd ‘𝐴)) → ∃𝑣 ∈ (2nd ‘𝐴)𝑣 <Q (*Q‘𝑞))
40		elprnqu 7839	. . . . . . . . . . . . . 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 481	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ P ∧ 𝑞 <Q 𝑟) ∧ 𝑣 ∈ (2nd ‘𝐴)) → 𝑣 ∈ Q)
43	42	ad2ant2r 513	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ P ∧ 𝑞 <Q 𝑟) ∧ (*Q‘𝑞) ∈ (2nd ‘𝐴)) ∧ (𝑣 ∈ (2nd ‘𝐴) ∧ 𝑣 <Q (*Q‘𝑞))) → 𝑣 ∈ Q)
44		recrecnq 7751	. . . . . . . . . . 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 537	. . . . . . . . . 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 4147	. . . . . . . . 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 492	. . . . . . . . . . 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 7749	. . . . . . . . . . 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 7777	. . . . . . . . . 10 ⊢ ((𝑞 ∈ Q ∧ (*Q‘𝑣) ∈ Q) → (𝑞 <Q (*Q‘𝑣) ↔ (*Q‘(*Q‘𝑣)) <Q (*Q‘𝑞)))
53	49, 51, 52	syl2anc 415	. . . . . . . . 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 535	. . . . . . . . 9 ⊢ ((((𝐴 ∈ P ∧ 𝑞 <Q 𝑟) ∧ (*Q‘𝑞) ∈ (2nd ‘𝐴)) ∧ (𝑣 ∈ (2nd ‘𝐴) ∧ 𝑣 <Q (*Q‘𝑞))) → 𝑣 ∈ (2nd ‘𝐴))
56	45, 55	eqeltrd 2315	. . . . . . . 8 ⊢ ((((𝐴 ∈ P ∧ 𝑞 <Q 𝑟) ∧ (*Q‘𝑞) ∈ (2nd ‘𝐴)) ∧ (𝑣 ∈ (2nd ‘𝐴) ∧ 𝑣 <Q (*Q‘𝑞))) → (*Q‘(*Q‘𝑣)) ∈ (2nd ‘𝐴))
57		breq2 4129	. . . . . . . . . . . 12 ⊢ (𝑦 = (*Q‘𝑣) → (𝑞 <Q 𝑦 ↔ 𝑞 <Q (*Q‘𝑣)))
58		fveq2 5690	. . . . . . . . . . . . 13 ⊢ (𝑦 = (*Q‘𝑣) → (*Q‘𝑦) = (*Q‘(*Q‘𝑣)))
59	58	eleq1d 2307	. . . . . . . . . . . 12 ⊢ (𝑦 = (*Q‘𝑣) → ((*Q‘𝑦) ∈ (2nd ‘𝐴) ↔ (*Q‘(*Q‘𝑣)) ∈ (2nd ‘𝐴)))
60	57, 59	anbi12d 477	. . . . . . . . . . 11 ⊢ (𝑦 = (*Q‘𝑣) → ((𝑞 <Q 𝑦 ∧ (*Q‘𝑦) ∈ (2nd ‘𝐴)) ↔ (𝑞 <Q (*Q‘𝑣) ∧ (*Q‘(*Q‘𝑣)) ∈ (2nd ‘𝐴))))
61	60	spcegv 2913	. . . . . . . . . 10 ⊢ ((*Q‘𝑣) ∈ Q → ((𝑞 <Q (*Q‘𝑣) ∧ (*Q‘(*Q‘𝑣)) ∈ (2nd ‘𝐴)) → ∃𝑦(𝑞 <Q 𝑦 ∧ (*Q‘𝑦) ∈ (2nd ‘𝐴))))
62	30	recexprlemell 7979	. . . . . . . . . 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 437	. . . . . . 7 ⊢ ((((𝐴 ∈ P ∧ 𝑞 <Q 𝑟) ∧ (*Q‘𝑞) ∈ (2nd ‘𝐴)) ∧ (𝑣 ∈ (2nd ‘𝐴) ∧ 𝑣 <Q (*Q‘𝑞))) → 𝑞 ∈ (1st ‘𝐵))
66	39, 65	rexlimddv 2673	. . . . . 6 ⊢ (((𝐴 ∈ P ∧ 𝑞 <Q 𝑟) ∧ (*Q‘𝑞) ∈ (2nd ‘𝐴)) → 𝑞 ∈ (1st ‘𝐵))
67	66	orcd 745	. . . . 5 ⊢ (((𝐴 ∈ P ∧ 𝑞 <Q 𝑟) ∧ (*Q‘𝑞) ∈ (2nd ‘𝐴)) → (𝑞 ∈ (1st ‘𝐵) ∨ 𝑟 ∈ (2nd ‘𝐵)))
68		ltrnqi 7778	. . . . . 6 ⊢ (𝑞 <Q 𝑟 → (*Q‘𝑟) <Q (*Q‘𝑞))
69		prloc 7848	. . . . . 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 810	. . . 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 2624	. 2 ⊢ (𝐴 ∈ P → ∀𝑟 ∈ Q (𝑞 <Q 𝑟 → (𝑞 ∈ (1st ‘𝐵) ∨ 𝑟 ∈ (2nd ‘𝐵))))
74	73	ralrimivw 2624	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 720   = wceq 1402  ∃wex 1545   ∈ wcel 2209  {cab 2224  ∀wral 2528  ∃wrex 2529  ⟨cop 3708   class class class wbr 4125  ‘cfv 5372  1^st c1st 6362  2^nd c2nd 6363  Qcnq 7637  *_Qcrq 7641   <_Q cltq 7642  Pcnp 7648

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4241  ax-sep 4244  ax-nul 4254  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-iinf 4730

This theorem depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-iun 4009  df-br 4126  df-opab 4188  df-mpt 4189  df-tr 4225  df-eprel 4429  df-id 4433  df-iord 4506  df-on 4508  df-suc 4511  df-iom 4733  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380  df-ov 6078  df-oprab 6079  df-mpo 6080  df-1st 6364  df-2nd 6365  df-recs 6566  df-irdg 6631  df-1o 6677  df-oadd 6681  df-omul 6682  df-er 6797  df-ec 6799  df-qs 6803  df-ni 7661  df-mi 7663  df-lti 7664  df-mpq 7702  df-enq 7704  df-nqqs 7705  df-mqqs 7707  df-1nqqs 7708  df-rq 7709  df-ltnqqs 7710  df-inp 7823

This theorem is referenced by:  recexprlempr  7989