Theorem recexprlemloc 6821

Description: 𝐵 is located. Lemma for recexpr 6828. (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 6665	. . . . . . . . 9 ⊢ (𝐴 ∈ P → ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P)
2		prnmaxl 6678	. . . . . . . . 9 ⊢ ((⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P ∧ (*Q‘𝑟) ∈ (1st ‘𝐴)) → ∃𝑢 ∈ (1st ‘𝐴)(*Q‘𝑟) <Q 𝑢)
3	1, 2	sylan 277	. . . . . . . 8 ⊢ ((𝐴 ∈ P ∧ (*Q‘𝑟) ∈ (1st ‘𝐴)) → ∃𝑢 ∈ (1st ‘𝐴)(*Q‘𝑟) <Q 𝑢)
4	3	adantlr 460	. . . . . . 7 ⊢ (((𝐴 ∈ P ∧ 𝑞 <Q 𝑟) ∧ (*Q‘𝑟) ∈ (1st ‘𝐴)) → ∃𝑢 ∈ (1st ‘𝐴)(*Q‘𝑟) <Q 𝑢)
5		simprr 498	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ P ∧ 𝑞 <Q 𝑟) ∧ (*Q‘𝑟) ∈ (1st ‘𝐴)) ∧ (𝑢 ∈ (1st ‘𝐴) ∧ (*Q‘𝑟) <Q 𝑢)) → (*Q‘𝑟) <Q 𝑢)
6		elprnql 6671	. . . . . . . . . . . . . 14 ⊢ ((⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P ∧ 𝑢 ∈ (1st ‘𝐴)) → 𝑢 ∈ Q)
7	1, 6	sylan 277	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ P ∧ 𝑢 ∈ (1st ‘𝐴)) → 𝑢 ∈ Q)
8	7	ad2ant2r 492	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ P ∧ 𝑞 <Q 𝑟) ∧ (𝑢 ∈ (1st ‘𝐴) ∧ (*Q‘𝑟) <Q 𝑢)) → 𝑢 ∈ Q)
9	8	adantlr 460	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ P ∧ 𝑞 <Q 𝑟) ∧ (*Q‘𝑟) ∈ (1st ‘𝐴)) ∧ (𝑢 ∈ (1st ‘𝐴) ∧ (*Q‘𝑟) <Q 𝑢)) → 𝑢 ∈ Q)
10		recrecnq 6584	. . . . . . . . . . 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 3811	. . . . . . . . 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 6582	. . . . . . . . . . 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 6555	. . . . . . . . . . . . . 14 ⊢ <Q ⊆ (Q × Q)
16	15	brel 4410	. . . . . . . . . . . . 13 ⊢ (𝑞 <Q 𝑟 → (𝑞 ∈ Q ∧ 𝑟 ∈ Q))
17	16	adantl 271	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ P ∧ 𝑞 <Q 𝑟) → (𝑞 ∈ Q ∧ 𝑟 ∈ Q))
18	17	ad2antrr 471	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ P ∧ 𝑞 <Q 𝑟) ∧ (*Q‘𝑟) ∈ (1st ‘𝐴)) ∧ (𝑢 ∈ (1st ‘𝐴) ∧ (*Q‘𝑟) <Q 𝑢)) → (𝑞 ∈ Q ∧ 𝑟 ∈ Q))
19	18	simprd 112	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ P ∧ 𝑞 <Q 𝑟) ∧ (*Q‘𝑟) ∈ (1st ‘𝐴)) ∧ (𝑢 ∈ (1st ‘𝐴) ∧ (*Q‘𝑟) <Q 𝑢)) → 𝑟 ∈ Q)
20		ltrnqg 6610	. . . . . . . . . 10 ⊢ (((*Q‘𝑢) ∈ Q ∧ 𝑟 ∈ Q) → ((*Q‘𝑢) <Q 𝑟 ↔ (*Q‘𝑟) <Q (*Q‘(*Q‘𝑢))))
21	14, 19, 20	syl2anc 403	. . . . . . . . 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 165	. . . . . . . 8 ⊢ ((((𝐴 ∈ P ∧ 𝑞 <Q 𝑟) ∧ (*Q‘𝑟) ∈ (1st ‘𝐴)) ∧ (𝑢 ∈ (1st ‘𝐴) ∧ (*Q‘𝑟) <Q 𝑢)) → (*Q‘𝑢) <Q 𝑟)
23		simprl 497	. . . . . . . . 9 ⊢ ((((𝐴 ∈ P ∧ 𝑞 <Q 𝑟) ∧ (*Q‘𝑟) ∈ (1st ‘𝐴)) ∧ (𝑢 ∈ (1st ‘𝐴) ∧ (*Q‘𝑟) <Q 𝑢)) → 𝑢 ∈ (1st ‘𝐴))
24	11, 23	eqeltrd 2155	. . . . . . . 8 ⊢ ((((𝐴 ∈ P ∧ 𝑞 <Q 𝑟) ∧ (*Q‘𝑟) ∈ (1st ‘𝐴)) ∧ (𝑢 ∈ (1st ‘𝐴) ∧ (*Q‘𝑟) <Q 𝑢)) → (*Q‘(*Q‘𝑢)) ∈ (1st ‘𝐴))
25		breq1 3788	. . . . . . . . . . . 12 ⊢ (𝑦 = (*Q‘𝑢) → (𝑦 <Q 𝑟 ↔ (*Q‘𝑢) <Q 𝑟))
26		fveq2 5198	. . . . . . . . . . . . 13 ⊢ (𝑦 = (*Q‘𝑢) → (*Q‘𝑦) = (*Q‘(*Q‘𝑢)))
27	26	eleq1d 2147	. . . . . . . . . . . 12 ⊢ (𝑦 = (*Q‘𝑢) → ((*Q‘𝑦) ∈ (1st ‘𝐴) ↔ (*Q‘(*Q‘𝑢)) ∈ (1st ‘𝐴)))
28	25, 27	anbi12d 456	. . . . . . . . . . 11 ⊢ (𝑦 = (*Q‘𝑢) → ((𝑦 <Q 𝑟 ∧ (*Q‘𝑦) ∈ (1st ‘𝐴)) ↔ ((*Q‘𝑢) <Q 𝑟 ∧ (*Q‘(*Q‘𝑢)) ∈ (1st ‘𝐴))))
29	28	spcegv 2686	. . . . . . . . . 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 6813	. . . . . . . . . 10 ⊢ (𝑟 ∈ (2nd ‘𝐵) ↔ ∃𝑦(𝑦 <Q 𝑟 ∧ (*Q‘𝑦) ∈ (1st ‘𝐴)))
32	29, 31	syl6ibr 160	. . . . . . . . 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 423	. . . . . . 7 ⊢ ((((𝐴 ∈ P ∧ 𝑞 <Q 𝑟) ∧ (*Q‘𝑟) ∈ (1st ‘𝐴)) ∧ (𝑢 ∈ (1st ‘𝐴) ∧ (*Q‘𝑟) <Q 𝑢)) → 𝑟 ∈ (2nd ‘𝐵))
35	4, 34	rexlimddv 2481	. . . . . 6 ⊢ (((𝐴 ∈ P ∧ 𝑞 <Q 𝑟) ∧ (*Q‘𝑟) ∈ (1st ‘𝐴)) → 𝑟 ∈ (2nd ‘𝐵))
36	35	olcd 685	. . . . 5 ⊢ (((𝐴 ∈ P ∧ 𝑞 <Q 𝑟) ∧ (*Q‘𝑟) ∈ (1st ‘𝐴)) → (𝑞 ∈ (1st ‘𝐵) ∨ 𝑟 ∈ (2nd ‘𝐵)))
37		prnminu 6679	. . . . . . . . 9 ⊢ ((⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P ∧ (*Q‘𝑞) ∈ (2nd ‘𝐴)) → ∃𝑣 ∈ (2nd ‘𝐴)𝑣 <Q (*Q‘𝑞))
38	1, 37	sylan 277	. . . . . . . 8 ⊢ ((𝐴 ∈ P ∧ (*Q‘𝑞) ∈ (2nd ‘𝐴)) → ∃𝑣 ∈ (2nd ‘𝐴)𝑣 <Q (*Q‘𝑞))
39	38	adantlr 460	. . . . . . 7 ⊢ (((𝐴 ∈ P ∧ 𝑞 <Q 𝑟) ∧ (*Q‘𝑞) ∈ (2nd ‘𝐴)) → ∃𝑣 ∈ (2nd ‘𝐴)𝑣 <Q (*Q‘𝑞))
40		elprnqu 6672	. . . . . . . . . . . . . 14 ⊢ ((⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P ∧ 𝑣 ∈ (2nd ‘𝐴)) → 𝑣 ∈ Q)
41	1, 40	sylan 277	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ P ∧ 𝑣 ∈ (2nd ‘𝐴)) → 𝑣 ∈ Q)
42	41	adantlr 460	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ P ∧ 𝑞 <Q 𝑟) ∧ 𝑣 ∈ (2nd ‘𝐴)) → 𝑣 ∈ Q)
43	42	ad2ant2r 492	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ P ∧ 𝑞 <Q 𝑟) ∧ (*Q‘𝑞) ∈ (2nd ‘𝐴)) ∧ (𝑣 ∈ (2nd ‘𝐴) ∧ 𝑣 <Q (*Q‘𝑞))) → 𝑣 ∈ Q)
44		recrecnq 6584	. . . . . . . . . . 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 498	. . . . . . . . . 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 3805	. . . . . . . . 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 471	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ P ∧ 𝑞 <Q 𝑟) ∧ (*Q‘𝑞) ∈ (2nd ‘𝐴)) ∧ (𝑣 ∈ (2nd ‘𝐴) ∧ 𝑣 <Q (*Q‘𝑞))) → (𝑞 ∈ Q ∧ 𝑟 ∈ Q))
49	48	simpld 110	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ P ∧ 𝑞 <Q 𝑟) ∧ (*Q‘𝑞) ∈ (2nd ‘𝐴)) ∧ (𝑣 ∈ (2nd ‘𝐴) ∧ 𝑣 <Q (*Q‘𝑞))) → 𝑞 ∈ Q)
50		recclnq 6582	. . . . . . . . . . 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 6610	. . . . . . . . . 10 ⊢ ((𝑞 ∈ Q ∧ (*Q‘𝑣) ∈ Q) → (𝑞 <Q (*Q‘𝑣) ↔ (*Q‘(*Q‘𝑣)) <Q (*Q‘𝑞)))
53	49, 51, 52	syl2anc 403	. . . . . . . . 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 165	. . . . . . . 8 ⊢ ((((𝐴 ∈ P ∧ 𝑞 <Q 𝑟) ∧ (*Q‘𝑞) ∈ (2nd ‘𝐴)) ∧ (𝑣 ∈ (2nd ‘𝐴) ∧ 𝑣 <Q (*Q‘𝑞))) → 𝑞 <Q (*Q‘𝑣))
55		simprl 497	. . . . . . . . 9 ⊢ ((((𝐴 ∈ P ∧ 𝑞 <Q 𝑟) ∧ (*Q‘𝑞) ∈ (2nd ‘𝐴)) ∧ (𝑣 ∈ (2nd ‘𝐴) ∧ 𝑣 <Q (*Q‘𝑞))) → 𝑣 ∈ (2nd ‘𝐴))
56	45, 55	eqeltrd 2155	. . . . . . . 8 ⊢ ((((𝐴 ∈ P ∧ 𝑞 <Q 𝑟) ∧ (*Q‘𝑞) ∈ (2nd ‘𝐴)) ∧ (𝑣 ∈ (2nd ‘𝐴) ∧ 𝑣 <Q (*Q‘𝑞))) → (*Q‘(*Q‘𝑣)) ∈ (2nd ‘𝐴))
57		breq2 3789	. . . . . . . . . . . 12 ⊢ (𝑦 = (*Q‘𝑣) → (𝑞 <Q 𝑦 ↔ 𝑞 <Q (*Q‘𝑣)))
58		fveq2 5198	. . . . . . . . . . . . 13 ⊢ (𝑦 = (*Q‘𝑣) → (*Q‘𝑦) = (*Q‘(*Q‘𝑣)))
59	58	eleq1d 2147	. . . . . . . . . . . 12 ⊢ (𝑦 = (*Q‘𝑣) → ((*Q‘𝑦) ∈ (2nd ‘𝐴) ↔ (*Q‘(*Q‘𝑣)) ∈ (2nd ‘𝐴)))
60	57, 59	anbi12d 456	. . . . . . . . . . 11 ⊢ (𝑦 = (*Q‘𝑣) → ((𝑞 <Q 𝑦 ∧ (*Q‘𝑦) ∈ (2nd ‘𝐴)) ↔ (𝑞 <Q (*Q‘𝑣) ∧ (*Q‘(*Q‘𝑣)) ∈ (2nd ‘𝐴))))
61	60	spcegv 2686	. . . . . . . . . 10 ⊢ ((*Q‘𝑣) ∈ Q → ((𝑞 <Q (*Q‘𝑣) ∧ (*Q‘(*Q‘𝑣)) ∈ (2nd ‘𝐴)) → ∃𝑦(𝑞 <Q 𝑦 ∧ (*Q‘𝑦) ∈ (2nd ‘𝐴))))
62	30	recexprlemell 6812	. . . . . . . . . 10 ⊢ (𝑞 ∈ (1st ‘𝐵) ↔ ∃𝑦(𝑞 <Q 𝑦 ∧ (*Q‘𝑦) ∈ (2nd ‘𝐴)))
63	61, 62	syl6ibr 160	. . . . . . . . 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 423	. . . . . . 7 ⊢ ((((𝐴 ∈ P ∧ 𝑞 <Q 𝑟) ∧ (*Q‘𝑞) ∈ (2nd ‘𝐴)) ∧ (𝑣 ∈ (2nd ‘𝐴) ∧ 𝑣 <Q (*Q‘𝑞))) → 𝑞 ∈ (1st ‘𝐵))
66	39, 65	rexlimddv 2481	. . . . . 6 ⊢ (((𝐴 ∈ P ∧ 𝑞 <Q 𝑟) ∧ (*Q‘𝑞) ∈ (2nd ‘𝐴)) → 𝑞 ∈ (1st ‘𝐵))
67	66	orcd 684	. . . . 5 ⊢ (((𝐴 ∈ P ∧ 𝑞 <Q 𝑟) ∧ (*Q‘𝑞) ∈ (2nd ‘𝐴)) → (𝑞 ∈ (1st ‘𝐵) ∨ 𝑟 ∈ (2nd ‘𝐵)))
68		ltrnqi 6611	. . . . . 6 ⊢ (𝑞 <Q 𝑟 → (*Q‘𝑟) <Q (*Q‘𝑞))
69		prloc 6681	. . . . . 6 ⊢ ((⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P ∧ (*Q‘𝑟) <Q (*Q‘𝑞)) → ((*Q‘𝑟) ∈ (1st ‘𝐴) ∨ (*Q‘𝑞) ∈ (2nd ‘𝐴)))
70	1, 68, 69	syl2an 283	. . . . 5 ⊢ ((𝐴 ∈ P ∧ 𝑞 <Q 𝑟) → ((*Q‘𝑟) ∈ (1st ‘𝐴) ∨ (*Q‘𝑞) ∈ (2nd ‘𝐴)))
71	36, 67, 70	mpjaodan 744	. . . 4 ⊢ ((𝐴 ∈ P ∧ 𝑞 <Q 𝑟) → (𝑞 ∈ (1st ‘𝐵) ∨ 𝑟 ∈ (2nd ‘𝐵)))
72	71	ex 113	. . 3 ⊢ (𝐴 ∈ P → (𝑞 <Q 𝑟 → (𝑞 ∈ (1st ‘𝐵) ∨ 𝑟 ∈ (2nd ‘𝐵))))
73	72	ralrimivw 2435	. 2 ⊢ (𝐴 ∈ P → ∀𝑟 ∈ Q (𝑞 <Q 𝑟 → (𝑞 ∈ (1st ‘𝐵) ∨ 𝑟 ∈ (2nd ‘𝐵))))
74	73	ralrimivw 2435	1 ⊢ (𝐴 ∈ P → ∀𝑞 ∈ Q ∀𝑟 ∈ Q (𝑞 <Q 𝑟 → (𝑞 ∈ (1st ‘𝐵) ∨ 𝑟 ∈ (2nd ‘𝐵))))

Syntax hints:   → wi 4   ∧ wa 102   ↔ wb 103   ∨ wo 661   = wceq 1284  ∃wex 1421   ∈ wcel 1433  {cab 2067  ∀wral 2348  ∃wrex 2349  ⟨cop 3401   class class class wbr 3785  ‘cfv 4922  1^st c1st 5785  2^nd c2nd 5786  Qcnq 6470  *_Qcrq 6474   <_Q cltq 6475  Pcnp 6481

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-13 1444  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-coll 3893  ax-sep 3896  ax-nul 3904  ax-pow 3948  ax-pr 3964  ax-un 4188  ax-setind 4280  ax-iinf 4329

This theorem depends on definitions:  df-bi 115  df-dc 776  df-3or 920  df-3an 921  df-tru 1287  df-fal 1290  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ne 2246  df-ral 2353  df-rex 2354  df-reu 2355  df-rab 2357  df-v 2603  df-sbc 2816  df-csb 2909  df-dif 2975  df-un 2977  df-in 2979  df-ss 2986  df-nul 3252  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-int 3637  df-iun 3680  df-br 3786  df-opab 3840  df-mpt 3841  df-tr 3876  df-eprel 4044  df-id 4048  df-iord 4121  df-on 4123  df-suc 4126  df-iom 4332  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-rn 4374  df-res 4375  df-ima 4376  df-iota 4887  df-fun 4924  df-fn 4925  df-f 4926  df-f1 4927  df-fo 4928  df-f1o 4929  df-fv 4930  df-ov 5535  df-oprab 5536  df-mpt2 5537  df-1st 5787  df-2nd 5788  df-recs 5943  df-irdg 5980  df-1o 6024  df-oadd 6028  df-omul 6029  df-er 6129  df-ec 6131  df-qs 6135  df-ni 6494  df-mi 6496  df-lti 6497  df-mpq 6535  df-enq 6537  df-nqqs 6538  df-mqqs 6540  df-1nqqs 6541  df-rq 6542  df-ltnqqs 6543  df-inp 6656

This theorem is referenced by:  recexprlempr  6822