Theorem recexprlemloc 7432

Description: 𝐵 is located. Lemma for recexpr 7439. (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 7276	. . . . . . . . 9 ⊢ (𝐴 ∈ P → ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P)
2		prnmaxl 7289	. . . . . . . . 9 ⊢ ((⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P ∧ (*Q‘𝑟) ∈ (1st ‘𝐴)) → ∃𝑢 ∈ (1st ‘𝐴)(*Q‘𝑟) <Q 𝑢)
3	1, 2	sylan 281	. . . . . . . 8 ⊢ ((𝐴 ∈ P ∧ (*Q‘𝑟) ∈ (1st ‘𝐴)) → ∃𝑢 ∈ (1st ‘𝐴)(*Q‘𝑟) <Q 𝑢)
4	3	adantlr 468	. . . . . . 7 ⊢ (((𝐴 ∈ P ∧ 𝑞 <Q 𝑟) ∧ (*Q‘𝑟) ∈ (1st ‘𝐴)) → ∃𝑢 ∈ (1st ‘𝐴)(*Q‘𝑟) <Q 𝑢)
5		simprr 521	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ P ∧ 𝑞 <Q 𝑟) ∧ (*Q‘𝑟) ∈ (1st ‘𝐴)) ∧ (𝑢 ∈ (1st ‘𝐴) ∧ (*Q‘𝑟) <Q 𝑢)) → (*Q‘𝑟) <Q 𝑢)
6		elprnql 7282	. . . . . . . . . . . . . 14 ⊢ ((⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P ∧ 𝑢 ∈ (1st ‘𝐴)) → 𝑢 ∈ Q)
7	1, 6	sylan 281	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ P ∧ 𝑢 ∈ (1st ‘𝐴)) → 𝑢 ∈ Q)
8	7	ad2ant2r 500	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ P ∧ 𝑞 <Q 𝑟) ∧ (𝑢 ∈ (1st ‘𝐴) ∧ (*Q‘𝑟) <Q 𝑢)) → 𝑢 ∈ Q)
9	8	adantlr 468	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ P ∧ 𝑞 <Q 𝑟) ∧ (*Q‘𝑟) ∈ (1st ‘𝐴)) ∧ (𝑢 ∈ (1st ‘𝐴) ∧ (*Q‘𝑟) <Q 𝑢)) → 𝑢 ∈ Q)
10		recrecnq 7195	. . . . . . . . . . 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 3951	. . . . . . . . 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 7193	. . . . . . . . . . 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 7166	. . . . . . . . . . . . . 14 ⊢ <Q ⊆ (Q × Q)
16	15	brel 4586	. . . . . . . . . . . . 13 ⊢ (𝑞 <Q 𝑟 → (𝑞 ∈ Q ∧ 𝑟 ∈ Q))
17	16	adantl 275	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ P ∧ 𝑞 <Q 𝑟) → (𝑞 ∈ Q ∧ 𝑟 ∈ Q))
18	17	ad2antrr 479	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ P ∧ 𝑞 <Q 𝑟) ∧ (*Q‘𝑟) ∈ (1st ‘𝐴)) ∧ (𝑢 ∈ (1st ‘𝐴) ∧ (*Q‘𝑟) <Q 𝑢)) → (𝑞 ∈ Q ∧ 𝑟 ∈ Q))
19	18	simprd 113	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ P ∧ 𝑞 <Q 𝑟) ∧ (*Q‘𝑟) ∈ (1st ‘𝐴)) ∧ (𝑢 ∈ (1st ‘𝐴) ∧ (*Q‘𝑟) <Q 𝑢)) → 𝑟 ∈ Q)
20		ltrnqg 7221	. . . . . . . . . 10 ⊢ (((*Q‘𝑢) ∈ Q ∧ 𝑟 ∈ Q) → ((*Q‘𝑢) <Q 𝑟 ↔ (*Q‘𝑟) <Q (*Q‘(*Q‘𝑢))))
21	14, 19, 20	syl2anc 408	. . . . . . . . 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 166	. . . . . . . 8 ⊢ ((((𝐴 ∈ P ∧ 𝑞 <Q 𝑟) ∧ (*Q‘𝑟) ∈ (1st ‘𝐴)) ∧ (𝑢 ∈ (1st ‘𝐴) ∧ (*Q‘𝑟) <Q 𝑢)) → (*Q‘𝑢) <Q 𝑟)
23		simprl 520	. . . . . . . . 9 ⊢ ((((𝐴 ∈ P ∧ 𝑞 <Q 𝑟) ∧ (*Q‘𝑟) ∈ (1st ‘𝐴)) ∧ (𝑢 ∈ (1st ‘𝐴) ∧ (*Q‘𝑟) <Q 𝑢)) → 𝑢 ∈ (1st ‘𝐴))
24	11, 23	eqeltrd 2214	. . . . . . . 8 ⊢ ((((𝐴 ∈ P ∧ 𝑞 <Q 𝑟) ∧ (*Q‘𝑟) ∈ (1st ‘𝐴)) ∧ (𝑢 ∈ (1st ‘𝐴) ∧ (*Q‘𝑟) <Q 𝑢)) → (*Q‘(*Q‘𝑢)) ∈ (1st ‘𝐴))
25		breq1 3927	. . . . . . . . . . . 12 ⊢ (𝑦 = (*Q‘𝑢) → (𝑦 <Q 𝑟 ↔ (*Q‘𝑢) <Q 𝑟))
26		fveq2 5414	. . . . . . . . . . . . 13 ⊢ (𝑦 = (*Q‘𝑢) → (*Q‘𝑦) = (*Q‘(*Q‘𝑢)))
27	26	eleq1d 2206	. . . . . . . . . . . 12 ⊢ (𝑦 = (*Q‘𝑢) → ((*Q‘𝑦) ∈ (1st ‘𝐴) ↔ (*Q‘(*Q‘𝑢)) ∈ (1st ‘𝐴)))
28	25, 27	anbi12d 464	. . . . . . . . . . 11 ⊢ (𝑦 = (*Q‘𝑢) → ((𝑦 <Q 𝑟 ∧ (*Q‘𝑦) ∈ (1st ‘𝐴)) ↔ ((*Q‘𝑢) <Q 𝑟 ∧ (*Q‘(*Q‘𝑢)) ∈ (1st ‘𝐴))))
29	28	spcegv 2769	. . . . . . . . . 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 7424	. . . . . . . . . 10 ⊢ (𝑟 ∈ (2nd ‘𝐵) ↔ ∃𝑦(𝑦 <Q 𝑟 ∧ (*Q‘𝑦) ∈ (1st ‘𝐴)))
32	29, 31	syl6ibr 161	. . . . . . . . 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 429	. . . . . . 7 ⊢ ((((𝐴 ∈ P ∧ 𝑞 <Q 𝑟) ∧ (*Q‘𝑟) ∈ (1st ‘𝐴)) ∧ (𝑢 ∈ (1st ‘𝐴) ∧ (*Q‘𝑟) <Q 𝑢)) → 𝑟 ∈ (2nd ‘𝐵))
35	4, 34	rexlimddv 2552	. . . . . 6 ⊢ (((𝐴 ∈ P ∧ 𝑞 <Q 𝑟) ∧ (*Q‘𝑟) ∈ (1st ‘𝐴)) → 𝑟 ∈ (2nd ‘𝐵))
36	35	olcd 723	. . . . 5 ⊢ (((𝐴 ∈ P ∧ 𝑞 <Q 𝑟) ∧ (*Q‘𝑟) ∈ (1st ‘𝐴)) → (𝑞 ∈ (1st ‘𝐵) ∨ 𝑟 ∈ (2nd ‘𝐵)))
37		prnminu 7290	. . . . . . . . 9 ⊢ ((⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P ∧ (*Q‘𝑞) ∈ (2nd ‘𝐴)) → ∃𝑣 ∈ (2nd ‘𝐴)𝑣 <Q (*Q‘𝑞))
38	1, 37	sylan 281	. . . . . . . 8 ⊢ ((𝐴 ∈ P ∧ (*Q‘𝑞) ∈ (2nd ‘𝐴)) → ∃𝑣 ∈ (2nd ‘𝐴)𝑣 <Q (*Q‘𝑞))
39	38	adantlr 468	. . . . . . 7 ⊢ (((𝐴 ∈ P ∧ 𝑞 <Q 𝑟) ∧ (*Q‘𝑞) ∈ (2nd ‘𝐴)) → ∃𝑣 ∈ (2nd ‘𝐴)𝑣 <Q (*Q‘𝑞))
40		elprnqu 7283	. . . . . . . . . . . . . 14 ⊢ ((⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P ∧ 𝑣 ∈ (2nd ‘𝐴)) → 𝑣 ∈ Q)
41	1, 40	sylan 281	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ P ∧ 𝑣 ∈ (2nd ‘𝐴)) → 𝑣 ∈ Q)
42	41	adantlr 468	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ P ∧ 𝑞 <Q 𝑟) ∧ 𝑣 ∈ (2nd ‘𝐴)) → 𝑣 ∈ Q)
43	42	ad2ant2r 500	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ P ∧ 𝑞 <Q 𝑟) ∧ (*Q‘𝑞) ∈ (2nd ‘𝐴)) ∧ (𝑣 ∈ (2nd ‘𝐴) ∧ 𝑣 <Q (*Q‘𝑞))) → 𝑣 ∈ Q)
44		recrecnq 7195	. . . . . . . . . . 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 521	. . . . . . . . . 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 3945	. . . . . . . . 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 479	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ P ∧ 𝑞 <Q 𝑟) ∧ (*Q‘𝑞) ∈ (2nd ‘𝐴)) ∧ (𝑣 ∈ (2nd ‘𝐴) ∧ 𝑣 <Q (*Q‘𝑞))) → (𝑞 ∈ Q ∧ 𝑟 ∈ Q))
49	48	simpld 111	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ P ∧ 𝑞 <Q 𝑟) ∧ (*Q‘𝑞) ∈ (2nd ‘𝐴)) ∧ (𝑣 ∈ (2nd ‘𝐴) ∧ 𝑣 <Q (*Q‘𝑞))) → 𝑞 ∈ Q)
50		recclnq 7193	. . . . . . . . . . 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 7221	. . . . . . . . . 10 ⊢ ((𝑞 ∈ Q ∧ (*Q‘𝑣) ∈ Q) → (𝑞 <Q (*Q‘𝑣) ↔ (*Q‘(*Q‘𝑣)) <Q (*Q‘𝑞)))
53	49, 51, 52	syl2anc 408	. . . . . . . . 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 166	. . . . . . . 8 ⊢ ((((𝐴 ∈ P ∧ 𝑞 <Q 𝑟) ∧ (*Q‘𝑞) ∈ (2nd ‘𝐴)) ∧ (𝑣 ∈ (2nd ‘𝐴) ∧ 𝑣 <Q (*Q‘𝑞))) → 𝑞 <Q (*Q‘𝑣))
55		simprl 520	. . . . . . . . 9 ⊢ ((((𝐴 ∈ P ∧ 𝑞 <Q 𝑟) ∧ (*Q‘𝑞) ∈ (2nd ‘𝐴)) ∧ (𝑣 ∈ (2nd ‘𝐴) ∧ 𝑣 <Q (*Q‘𝑞))) → 𝑣 ∈ (2nd ‘𝐴))
56	45, 55	eqeltrd 2214	. . . . . . . 8 ⊢ ((((𝐴 ∈ P ∧ 𝑞 <Q 𝑟) ∧ (*Q‘𝑞) ∈ (2nd ‘𝐴)) ∧ (𝑣 ∈ (2nd ‘𝐴) ∧ 𝑣 <Q (*Q‘𝑞))) → (*Q‘(*Q‘𝑣)) ∈ (2nd ‘𝐴))
57		breq2 3928	. . . . . . . . . . . 12 ⊢ (𝑦 = (*Q‘𝑣) → (𝑞 <Q 𝑦 ↔ 𝑞 <Q (*Q‘𝑣)))
58		fveq2 5414	. . . . . . . . . . . . 13 ⊢ (𝑦 = (*Q‘𝑣) → (*Q‘𝑦) = (*Q‘(*Q‘𝑣)))
59	58	eleq1d 2206	. . . . . . . . . . . 12 ⊢ (𝑦 = (*Q‘𝑣) → ((*Q‘𝑦) ∈ (2nd ‘𝐴) ↔ (*Q‘(*Q‘𝑣)) ∈ (2nd ‘𝐴)))
60	57, 59	anbi12d 464	. . . . . . . . . . 11 ⊢ (𝑦 = (*Q‘𝑣) → ((𝑞 <Q 𝑦 ∧ (*Q‘𝑦) ∈ (2nd ‘𝐴)) ↔ (𝑞 <Q (*Q‘𝑣) ∧ (*Q‘(*Q‘𝑣)) ∈ (2nd ‘𝐴))))
61	60	spcegv 2769	. . . . . . . . . 10 ⊢ ((*Q‘𝑣) ∈ Q → ((𝑞 <Q (*Q‘𝑣) ∧ (*Q‘(*Q‘𝑣)) ∈ (2nd ‘𝐴)) → ∃𝑦(𝑞 <Q 𝑦 ∧ (*Q‘𝑦) ∈ (2nd ‘𝐴))))
62	30	recexprlemell 7423	. . . . . . . . . 10 ⊢ (𝑞 ∈ (1st ‘𝐵) ↔ ∃𝑦(𝑞 <Q 𝑦 ∧ (*Q‘𝑦) ∈ (2nd ‘𝐴)))
63	61, 62	syl6ibr 161	. . . . . . . . 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 429	. . . . . . 7 ⊢ ((((𝐴 ∈ P ∧ 𝑞 <Q 𝑟) ∧ (*Q‘𝑞) ∈ (2nd ‘𝐴)) ∧ (𝑣 ∈ (2nd ‘𝐴) ∧ 𝑣 <Q (*Q‘𝑞))) → 𝑞 ∈ (1st ‘𝐵))
66	39, 65	rexlimddv 2552	. . . . . 6 ⊢ (((𝐴 ∈ P ∧ 𝑞 <Q 𝑟) ∧ (*Q‘𝑞) ∈ (2nd ‘𝐴)) → 𝑞 ∈ (1st ‘𝐵))
67	66	orcd 722	. . . . 5 ⊢ (((𝐴 ∈ P ∧ 𝑞 <Q 𝑟) ∧ (*Q‘𝑞) ∈ (2nd ‘𝐴)) → (𝑞 ∈ (1st ‘𝐵) ∨ 𝑟 ∈ (2nd ‘𝐵)))
68		ltrnqi 7222	. . . . . 6 ⊢ (𝑞 <Q 𝑟 → (*Q‘𝑟) <Q (*Q‘𝑞))
69		prloc 7292	. . . . . 6 ⊢ ((⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P ∧ (*Q‘𝑟) <Q (*Q‘𝑞)) → ((*Q‘𝑟) ∈ (1st ‘𝐴) ∨ (*Q‘𝑞) ∈ (2nd ‘𝐴)))
70	1, 68, 69	syl2an 287	. . . . 5 ⊢ ((𝐴 ∈ P ∧ 𝑞 <Q 𝑟) → ((*Q‘𝑟) ∈ (1st ‘𝐴) ∨ (*Q‘𝑞) ∈ (2nd ‘𝐴)))
71	36, 67, 70	mpjaodan 787	. . . 4 ⊢ ((𝐴 ∈ P ∧ 𝑞 <Q 𝑟) → (𝑞 ∈ (1st ‘𝐵) ∨ 𝑟 ∈ (2nd ‘𝐵)))
72	71	ex 114	. . 3 ⊢ (𝐴 ∈ P → (𝑞 <Q 𝑟 → (𝑞 ∈ (1st ‘𝐵) ∨ 𝑟 ∈ (2nd ‘𝐵))))
73	72	ralrimivw 2504	. 2 ⊢ (𝐴 ∈ P → ∀𝑟 ∈ Q (𝑞 <Q 𝑟 → (𝑞 ∈ (1st ‘𝐵) ∨ 𝑟 ∈ (2nd ‘𝐵))))
74	73	ralrimivw 2504	1 ⊢ (𝐴 ∈ P → ∀𝑞 ∈ Q ∀𝑟 ∈ Q (𝑞 <Q 𝑟 → (𝑞 ∈ (1st ‘𝐵) ∨ 𝑟 ∈ (2nd ‘𝐵))))

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   ∨ wo 697   = wceq 1331  ∃wex 1468   ∈ wcel 1480  {cab 2123  ∀wral 2414  ∃wrex 2415  ⟨cop 3525   class class class wbr 3924  ‘cfv 5118  1^st c1st 6029  2^nd c2nd 6030  Qcnq 7081  *_Qcrq 7085   <_Q cltq 7086  Pcnp 7092

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 2119  ax-coll 4038  ax-sep 4041  ax-nul 4049  ax-pow 4093  ax-pr 4126  ax-un 4350  ax-setind 4447  ax-iinf 4497

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 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ne 2307  df-ral 2419  df-rex 2420  df-reu 2421  df-rab 2423  df-v 2683  df-sbc 2905  df-csb 2999  df-dif 3068  df-un 3070  df-in 3072  df-ss 3079  df-nul 3359  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-int 3767  df-iun 3810  df-br 3925  df-opab 3985  df-mpt 3986  df-tr 4022  df-eprel 4206  df-id 4210  df-iord 4283  df-on 4285  df-suc 4288  df-iom 4500  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-rn 4545  df-res 4546  df-ima 4547  df-iota 5083  df-fun 5120  df-fn 5121  df-f 5122  df-f1 5123  df-fo 5124  df-f1o 5125  df-fv 5126  df-ov 5770  df-oprab 5771  df-mpo 5772  df-1st 6031  df-2nd 6032  df-recs 6195  df-irdg 6260  df-1o 6306  df-oadd 6310  df-omul 6311  df-er 6422  df-ec 6424  df-qs 6428  df-ni 7105  df-mi 7107  df-lti 7108  df-mpq 7146  df-enq 7148  df-nqqs 7149  df-mqqs 7151  df-1nqqs 7152  df-rq 7153  df-ltnqqs 7154  df-inp 7267

This theorem is referenced by:  recexprlempr  7433