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Theorem ltexprlemdisj 7262

Description: Our constructed difference is disjoint. Lemma for ltexpri 7269. (Contributed by Jim Kingdon, 17-Dec-2019.)

**Hypothesis**
Ref	Expression
ltexprlem.1	⊢ 𝐶 = ⟨{𝑥 ∈ Q ∣ ∃𝑦(𝑦 ∈ (2^nd ‘𝐴) ∧ (𝑦 +_Q 𝑥) ∈ (1^st ‘𝐵))}, {𝑥 ∈ Q ∣ ∃𝑦(𝑦 ∈ (1^st ‘𝐴) ∧ (𝑦 +_Q 𝑥) ∈ (2^nd ‘𝐵))}⟩

**Assertion**
Ref	Expression
ltexprlemdisj	⊢ (𝐴<_P 𝐵 → ∀𝑞 ∈ Q ¬ (𝑞 ∈ (1^st ‘𝐶) ∧ 𝑞 ∈ (2^nd ‘𝐶)))

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

Proof of Theorem ltexprlemdisj Dummy variables 𝑧 𝑓 𝑔 ℎ are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ltsonq 7054	. . . . . 6 ⊢ <_Q Or Q
2		ltrelnq 7021	. . . . . 6 ⊢ <_Q ⊆ (Q × Q)
3	1, 2	son2lpi 4861	. . . . 5 ⊢ ¬ (𝑦 <_Q 𝑧 ∧ 𝑧 <_Q 𝑦)
4		ltrelpr 7161	. . . . . . . . . . . . . . . 16 ⊢ <_P ⊆ (P × P)
5	4	brel 4519	. . . . . . . . . . . . . . 15 ⊢ (𝐴<_P 𝐵 → (𝐴 ∈ P ∧ 𝐵 ∈ P))
6	5	simprd 113	. . . . . . . . . . . . . 14 ⊢ (𝐴<_P 𝐵 → 𝐵 ∈ P)
7		prop 7131	. . . . . . . . . . . . . 14 ⊢ (𝐵 ∈ P → ⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩ ∈ P)
8	6, 7	syl 14	. . . . . . . . . . . . 13 ⊢ (𝐴<_P 𝐵 → ⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩ ∈ P)
9		prltlu 7143	. . . . . . . . . . . . 13 ⊢ ((⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩ ∈ P ∧ (𝑦 +_Q 𝑞) ∈ (1^st ‘𝐵) ∧ (𝑧 +_Q 𝑞) ∈ (2^nd ‘𝐵)) → (𝑦 +_Q 𝑞) <_Q (𝑧 +_Q 𝑞))
10	8, 9	syl3an1 1214	. . . . . . . . . . . 12 ⊢ ((𝐴<_P 𝐵 ∧ (𝑦 +_Q 𝑞) ∈ (1^st ‘𝐵) ∧ (𝑧 +_Q 𝑞) ∈ (2^nd ‘𝐵)) → (𝑦 +_Q 𝑞) <_Q (𝑧 +_Q 𝑞))
11	10	3expb 1147	. . . . . . . . . . 11 ⊢ ((𝐴<_P 𝐵 ∧ ((𝑦 +_Q 𝑞) ∈ (1^st ‘𝐵) ∧ (𝑧 +_Q 𝑞) ∈ (2^nd ‘𝐵))) → (𝑦 +_Q 𝑞) <_Q (𝑧 +_Q 𝑞))
12	11	adantlr 462	. . . . . . . . . 10 ⊢ (((𝐴<_P 𝐵 ∧ 𝑞 ∈ Q) ∧ ((𝑦 +_Q 𝑞) ∈ (1^st ‘𝐵) ∧ (𝑧 +_Q 𝑞) ∈ (2^nd ‘𝐵))) → (𝑦 +_Q 𝑞) <_Q (𝑧 +_Q 𝑞))
13	12	adantrll 469	. . . . . . . . 9 ⊢ (((𝐴<_P 𝐵 ∧ 𝑞 ∈ Q) ∧ ((𝑦 ∈ (2^nd ‘𝐴) ∧ (𝑦 +_Q 𝑞) ∈ (1^st ‘𝐵)) ∧ (𝑧 +_Q 𝑞) ∈ (2^nd ‘𝐵))) → (𝑦 +_Q 𝑞) <_Q (𝑧 +_Q 𝑞))
14	13	adantrrl 471	. . . . . . . 8 ⊢ (((𝐴<_P 𝐵 ∧ 𝑞 ∈ Q) ∧ ((𝑦 ∈ (2^nd ‘𝐴) ∧ (𝑦 +_Q 𝑞) ∈ (1^st ‘𝐵)) ∧ (𝑧 ∈ (1^st ‘𝐴) ∧ (𝑧 +_Q 𝑞) ∈ (2^nd ‘𝐵)))) → (𝑦 +_Q 𝑞) <_Q (𝑧 +_Q 𝑞))
15		ltanqg 7056	. . . . . . . . . 10 ⊢ ((𝑓 ∈ Q ∧ 𝑔 ∈ Q ∧ ℎ ∈ Q) → (𝑓 <_Q 𝑔 ↔ (ℎ +_Q 𝑓) <_Q (ℎ +_Q 𝑔)))
16	15	adantl 272	. . . . . . . . 9 ⊢ ((((𝐴<_P 𝐵 ∧ 𝑞 ∈ Q) ∧ ((𝑦 ∈ (2^nd ‘𝐴) ∧ (𝑦 +_Q 𝑞) ∈ (1^st ‘𝐵)) ∧ (𝑧 ∈ (1^st ‘𝐴) ∧ (𝑧 +_Q 𝑞) ∈ (2^nd ‘𝐵)))) ∧ (𝑓 ∈ Q ∧ 𝑔 ∈ Q ∧ ℎ ∈ Q)) → (𝑓 <_Q 𝑔 ↔ (ℎ +_Q 𝑓) <_Q (ℎ +_Q 𝑔)))
17	5	simpld 111	. . . . . . . . . . . . 13 ⊢ (𝐴<_P 𝐵 → 𝐴 ∈ P)
18		prop 7131	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ P → ⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩ ∈ P)
19	17, 18	syl 14	. . . . . . . . . . . 12 ⊢ (𝐴<_P 𝐵 → ⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩ ∈ P)
20		elprnqu 7138	. . . . . . . . . . . 12 ⊢ ((⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩ ∈ P ∧ 𝑦 ∈ (2^nd ‘𝐴)) → 𝑦 ∈ Q)
21	19, 20	sylan 278	. . . . . . . . . . 11 ⊢ ((𝐴<_P 𝐵 ∧ 𝑦 ∈ (2^nd ‘𝐴)) → 𝑦 ∈ Q)
22	21	ad2ant2r 494	. . . . . . . . . 10 ⊢ (((𝐴<_P 𝐵 ∧ 𝑞 ∈ Q) ∧ (𝑦 ∈ (2^nd ‘𝐴) ∧ (𝑦 +_Q 𝑞) ∈ (1^st ‘𝐵))) → 𝑦 ∈ Q)
23	22	adantrr 464	. . . . . . . . 9 ⊢ (((𝐴<_P 𝐵 ∧ 𝑞 ∈ Q) ∧ ((𝑦 ∈ (2^nd ‘𝐴) ∧ (𝑦 +_Q 𝑞) ∈ (1^st ‘𝐵)) ∧ (𝑧 ∈ (1^st ‘𝐴) ∧ (𝑧 +_Q 𝑞) ∈ (2^nd ‘𝐵)))) → 𝑦 ∈ Q)
24		elprnql 7137	. . . . . . . . . . . 12 ⊢ ((⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩ ∈ P ∧ 𝑧 ∈ (1^st ‘𝐴)) → 𝑧 ∈ Q)
25	19, 24	sylan 278	. . . . . . . . . . 11 ⊢ ((𝐴<_P 𝐵 ∧ 𝑧 ∈ (1^st ‘𝐴)) → 𝑧 ∈ Q)
26	25	ad2ant2r 494	. . . . . . . . . 10 ⊢ (((𝐴<_P 𝐵 ∧ 𝑞 ∈ Q) ∧ (𝑧 ∈ (1^st ‘𝐴) ∧ (𝑧 +_Q 𝑞) ∈ (2^nd ‘𝐵))) → 𝑧 ∈ Q)
27	26	adantrl 463	. . . . . . . . 9 ⊢ (((𝐴<_P 𝐵 ∧ 𝑞 ∈ Q) ∧ ((𝑦 ∈ (2^nd ‘𝐴) ∧ (𝑦 +_Q 𝑞) ∈ (1^st ‘𝐵)) ∧ (𝑧 ∈ (1^st ‘𝐴) ∧ (𝑧 +_Q 𝑞) ∈ (2^nd ‘𝐵)))) → 𝑧 ∈ Q)
28		simplr 498	. . . . . . . . 9 ⊢ (((𝐴<_P 𝐵 ∧ 𝑞 ∈ Q) ∧ ((𝑦 ∈ (2^nd ‘𝐴) ∧ (𝑦 +_Q 𝑞) ∈ (1^st ‘𝐵)) ∧ (𝑧 ∈ (1^st ‘𝐴) ∧ (𝑧 +_Q 𝑞) ∈ (2^nd ‘𝐵)))) → 𝑞 ∈ Q)
29		addcomnqg 7037	. . . . . . . . . 10 ⊢ ((𝑓 ∈ Q ∧ 𝑔 ∈ Q) → (𝑓 +_Q 𝑔) = (𝑔 +_Q 𝑓))
30	29	adantl 272	. . . . . . . . 9 ⊢ ((((𝐴<_P 𝐵 ∧ 𝑞 ∈ Q) ∧ ((𝑦 ∈ (2^nd ‘𝐴) ∧ (𝑦 +_Q 𝑞) ∈ (1^st ‘𝐵)) ∧ (𝑧 ∈ (1^st ‘𝐴) ∧ (𝑧 +_Q 𝑞) ∈ (2^nd ‘𝐵)))) ∧ (𝑓 ∈ Q ∧ 𝑔 ∈ Q)) → (𝑓 +_Q 𝑔) = (𝑔 +_Q 𝑓))
31	16, 23, 27, 28, 30	caovord2d 5852	. . . . . . . 8 ⊢ (((𝐴<_P 𝐵 ∧ 𝑞 ∈ Q) ∧ ((𝑦 ∈ (2^nd ‘𝐴) ∧ (𝑦 +_Q 𝑞) ∈ (1^st ‘𝐵)) ∧ (𝑧 ∈ (1^st ‘𝐴) ∧ (𝑧 +_Q 𝑞) ∈ (2^nd ‘𝐵)))) → (𝑦 <_Q 𝑧 ↔ (𝑦 +_Q 𝑞) <_Q (𝑧 +_Q 𝑞)))
32	14, 31	mpbird 166	. . . . . . 7 ⊢ (((𝐴<_P 𝐵 ∧ 𝑞 ∈ Q) ∧ ((𝑦 ∈ (2^nd ‘𝐴) ∧ (𝑦 +_Q 𝑞) ∈ (1^st ‘𝐵)) ∧ (𝑧 ∈ (1^st ‘𝐴) ∧ (𝑧 +_Q 𝑞) ∈ (2^nd ‘𝐵)))) → 𝑦 <_Q 𝑧)
33		prltlu 7143	. . . . . . . . . . . . 13 ⊢ ((⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩ ∈ P ∧ 𝑧 ∈ (1^st ‘𝐴) ∧ 𝑦 ∈ (2^nd ‘𝐴)) → 𝑧 <_Q 𝑦)
34	19, 33	syl3an1 1214	. . . . . . . . . . . 12 ⊢ ((𝐴<_P 𝐵 ∧ 𝑧 ∈ (1^st ‘𝐴) ∧ 𝑦 ∈ (2^nd ‘𝐴)) → 𝑧 <_Q 𝑦)
35	34	3com23 1152	. . . . . . . . . . 11 ⊢ ((𝐴<_P 𝐵 ∧ 𝑦 ∈ (2^nd ‘𝐴) ∧ 𝑧 ∈ (1^st ‘𝐴)) → 𝑧 <_Q 𝑦)
36	35	3expb 1147	. . . . . . . . . 10 ⊢ ((𝐴<_P 𝐵 ∧ (𝑦 ∈ (2^nd ‘𝐴) ∧ 𝑧 ∈ (1^st ‘𝐴))) → 𝑧 <_Q 𝑦)
37	36	adantlr 462	. . . . . . . . 9 ⊢ (((𝐴<_P 𝐵 ∧ 𝑞 ∈ Q) ∧ (𝑦 ∈ (2^nd ‘𝐴) ∧ 𝑧 ∈ (1^st ‘𝐴))) → 𝑧 <_Q 𝑦)
38	37	adantrlr 470	. . . . . . . 8 ⊢ (((𝐴<_P 𝐵 ∧ 𝑞 ∈ Q) ∧ ((𝑦 ∈ (2^nd ‘𝐴) ∧ (𝑦 +_Q 𝑞) ∈ (1^st ‘𝐵)) ∧ 𝑧 ∈ (1^st ‘𝐴))) → 𝑧 <_Q 𝑦)
39	38	adantrrr 472	. . . . . . 7 ⊢ (((𝐴<_P 𝐵 ∧ 𝑞 ∈ Q) ∧ ((𝑦 ∈ (2^nd ‘𝐴) ∧ (𝑦 +_Q 𝑞) ∈ (1^st ‘𝐵)) ∧ (𝑧 ∈ (1^st ‘𝐴) ∧ (𝑧 +_Q 𝑞) ∈ (2^nd ‘𝐵)))) → 𝑧 <_Q 𝑦)
40	32, 39	jca 301	. . . . . 6 ⊢ (((𝐴<_P 𝐵 ∧ 𝑞 ∈ Q) ∧ ((𝑦 ∈ (2^nd ‘𝐴) ∧ (𝑦 +_Q 𝑞) ∈ (1^st ‘𝐵)) ∧ (𝑧 ∈ (1^st ‘𝐴) ∧ (𝑧 +_Q 𝑞) ∈ (2^nd ‘𝐵)))) → (𝑦 <_Q 𝑧 ∧ 𝑧 <_Q 𝑦))
41	40	ex 114	. . . . 5 ⊢ ((𝐴<_P 𝐵 ∧ 𝑞 ∈ Q) → (((𝑦 ∈ (2^nd ‘𝐴) ∧ (𝑦 +_Q 𝑞) ∈ (1^st ‘𝐵)) ∧ (𝑧 ∈ (1^st ‘𝐴) ∧ (𝑧 +_Q 𝑞) ∈ (2^nd ‘𝐵))) → (𝑦 <_Q 𝑧 ∧ 𝑧 <_Q 𝑦)))
42	3, 41	mtoi 628	. . . 4 ⊢ ((𝐴<_P 𝐵 ∧ 𝑞 ∈ Q) → ¬ ((𝑦 ∈ (2^nd ‘𝐴) ∧ (𝑦 +_Q 𝑞) ∈ (1^st ‘𝐵)) ∧ (𝑧 ∈ (1^st ‘𝐴) ∧ (𝑧 +_Q 𝑞) ∈ (2^nd ‘𝐵))))
43	42	alrimivv 1810	. . 3 ⊢ ((𝐴<_P 𝐵 ∧ 𝑞 ∈ Q) → ∀𝑦∀𝑧 ¬ ((𝑦 ∈ (2^nd ‘𝐴) ∧ (𝑦 +_Q 𝑞) ∈ (1^st ‘𝐵)) ∧ (𝑧 ∈ (1^st ‘𝐴) ∧ (𝑧 +_Q 𝑞) ∈ (2^nd ‘𝐵))))
44		ltexprlem.1	. . . . . . . . . . . 12 ⊢ 𝐶 = ⟨{𝑥 ∈ Q ∣ ∃𝑦(𝑦 ∈ (2^nd ‘𝐴) ∧ (𝑦 +_Q 𝑥) ∈ (1^st ‘𝐵))}, {𝑥 ∈ Q ∣ ∃𝑦(𝑦 ∈ (1^st ‘𝐴) ∧ (𝑦 +_Q 𝑥) ∈ (2^nd ‘𝐵))}⟩
45	44	ltexprlemell 7254	. . . . . . . . . . 11 ⊢ (𝑞 ∈ (1^st ‘𝐶) ↔ (𝑞 ∈ Q ∧ ∃𝑦(𝑦 ∈ (2^nd ‘𝐴) ∧ (𝑦 +_Q 𝑞) ∈ (1^st ‘𝐵))))
46	44	ltexprlemelu 7255	. . . . . . . . . . 11 ⊢ (𝑞 ∈ (2^nd ‘𝐶) ↔ (𝑞 ∈ Q ∧ ∃𝑦(𝑦 ∈ (1^st ‘𝐴) ∧ (𝑦 +_Q 𝑞) ∈ (2^nd ‘𝐵))))
47	45, 46	anbi12i 449	. . . . . . . . . 10 ⊢ ((𝑞 ∈ (1^st ‘𝐶) ∧ 𝑞 ∈ (2^nd ‘𝐶)) ↔ ((𝑞 ∈ Q ∧ ∃𝑦(𝑦 ∈ (2^nd ‘𝐴) ∧ (𝑦 +_Q 𝑞) ∈ (1^st ‘𝐵))) ∧ (𝑞 ∈ Q ∧ ∃𝑦(𝑦 ∈ (1^st ‘𝐴) ∧ (𝑦 +_Q 𝑞) ∈ (2^nd ‘𝐵)))))
48		anandi 558	. . . . . . . . . 10 ⊢ ((𝑞 ∈ Q ∧ (∃𝑦(𝑦 ∈ (2^nd ‘𝐴) ∧ (𝑦 +_Q 𝑞) ∈ (1^st ‘𝐵)) ∧ ∃𝑦(𝑦 ∈ (1^st ‘𝐴) ∧ (𝑦 +_Q 𝑞) ∈ (2^nd ‘𝐵)))) ↔ ((𝑞 ∈ Q ∧ ∃𝑦(𝑦 ∈ (2^nd ‘𝐴) ∧ (𝑦 +_Q 𝑞) ∈ (1^st ‘𝐵))) ∧ (𝑞 ∈ Q ∧ ∃𝑦(𝑦 ∈ (1^st ‘𝐴) ∧ (𝑦 +_Q 𝑞) ∈ (2^nd ‘𝐵)))))
49	47, 48	bitr4i 186	. . . . . . . . 9 ⊢ ((𝑞 ∈ (1^st ‘𝐶) ∧ 𝑞 ∈ (2^nd ‘𝐶)) ↔ (𝑞 ∈ Q ∧ (∃𝑦(𝑦 ∈ (2^nd ‘𝐴) ∧ (𝑦 +_Q 𝑞) ∈ (1^st ‘𝐵)) ∧ ∃𝑦(𝑦 ∈ (1^st ‘𝐴) ∧ (𝑦 +_Q 𝑞) ∈ (2^nd ‘𝐵)))))
50	49	baib 869	. . . . . . . 8 ⊢ (𝑞 ∈ Q → ((𝑞 ∈ (1^st ‘𝐶) ∧ 𝑞 ∈ (2^nd ‘𝐶)) ↔ (∃𝑦(𝑦 ∈ (2^nd ‘𝐴) ∧ (𝑦 +_Q 𝑞) ∈ (1^st ‘𝐵)) ∧ ∃𝑦(𝑦 ∈ (1^st ‘𝐴) ∧ (𝑦 +_Q 𝑞) ∈ (2^nd ‘𝐵)))))
51		eleq1 2157	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑧 → (𝑦 ∈ (1^st ‘𝐴) ↔ 𝑧 ∈ (1^st ‘𝐴)))
52		oveq1 5697	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑧 → (𝑦 +_Q 𝑞) = (𝑧 +_Q 𝑞))
53	52	eleq1d 2163	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑧 → ((𝑦 +_Q 𝑞) ∈ (2^nd ‘𝐵) ↔ (𝑧 +_Q 𝑞) ∈ (2^nd ‘𝐵)))
54	51, 53	anbi12d 458	. . . . . . . . . 10 ⊢ (𝑦 = 𝑧 → ((𝑦 ∈ (1^st ‘𝐴) ∧ (𝑦 +_Q 𝑞) ∈ (2^nd ‘𝐵)) ↔ (𝑧 ∈ (1^st ‘𝐴) ∧ (𝑧 +_Q 𝑞) ∈ (2^nd ‘𝐵))))
55	54	cbvexv 1850	. . . . . . . . 9 ⊢ (∃𝑦(𝑦 ∈ (1^st ‘𝐴) ∧ (𝑦 +_Q 𝑞) ∈ (2^nd ‘𝐵)) ↔ ∃𝑧(𝑧 ∈ (1^st ‘𝐴) ∧ (𝑧 +_Q 𝑞) ∈ (2^nd ‘𝐵)))
56	55	anbi2i 446	. . . . . . . 8 ⊢ ((∃𝑦(𝑦 ∈ (2^nd ‘𝐴) ∧ (𝑦 +_Q 𝑞) ∈ (1^st ‘𝐵)) ∧ ∃𝑦(𝑦 ∈ (1^st ‘𝐴) ∧ (𝑦 +_Q 𝑞) ∈ (2^nd ‘𝐵))) ↔ (∃𝑦(𝑦 ∈ (2^nd ‘𝐴) ∧ (𝑦 +_Q 𝑞) ∈ (1^st ‘𝐵)) ∧ ∃𝑧(𝑧 ∈ (1^st ‘𝐴) ∧ (𝑧 +_Q 𝑞) ∈ (2^nd ‘𝐵))))
57	50, 56	syl6bb 195	. . . . . . 7 ⊢ (𝑞 ∈ Q → ((𝑞 ∈ (1^st ‘𝐶) ∧ 𝑞 ∈ (2^nd ‘𝐶)) ↔ (∃𝑦(𝑦 ∈ (2^nd ‘𝐴) ∧ (𝑦 +_Q 𝑞) ∈ (1^st ‘𝐵)) ∧ ∃𝑧(𝑧 ∈ (1^st ‘𝐴) ∧ (𝑧 +_Q 𝑞) ∈ (2^nd ‘𝐵)))))
58		eeanv 1862	. . . . . . 7 ⊢ (∃𝑦∃𝑧((𝑦 ∈ (2^nd ‘𝐴) ∧ (𝑦 +_Q 𝑞) ∈ (1^st ‘𝐵)) ∧ (𝑧 ∈ (1^st ‘𝐴) ∧ (𝑧 +_Q 𝑞) ∈ (2^nd ‘𝐵))) ↔ (∃𝑦(𝑦 ∈ (2^nd ‘𝐴) ∧ (𝑦 +_Q 𝑞) ∈ (1^st ‘𝐵)) ∧ ∃𝑧(𝑧 ∈ (1^st ‘𝐴) ∧ (𝑧 +_Q 𝑞) ∈ (2^nd ‘𝐵))))
59	57, 58	syl6bbr 197	. . . . . 6 ⊢ (𝑞 ∈ Q → ((𝑞 ∈ (1^st ‘𝐶) ∧ 𝑞 ∈ (2^nd ‘𝐶)) ↔ ∃𝑦∃𝑧((𝑦 ∈ (2^nd ‘𝐴) ∧ (𝑦 +_Q 𝑞) ∈ (1^st ‘𝐵)) ∧ (𝑧 ∈ (1^st ‘𝐴) ∧ (𝑧 +_Q 𝑞) ∈ (2^nd ‘𝐵)))))
60	59	notbid 630	. . . . 5 ⊢ (𝑞 ∈ Q → (¬ (𝑞 ∈ (1^st ‘𝐶) ∧ 𝑞 ∈ (2^nd ‘𝐶)) ↔ ¬ ∃𝑦∃𝑧((𝑦 ∈ (2^nd ‘𝐴) ∧ (𝑦 +_Q 𝑞) ∈ (1^st ‘𝐵)) ∧ (𝑧 ∈ (1^st ‘𝐴) ∧ (𝑧 +_Q 𝑞) ∈ (2^nd ‘𝐵)))))
61		alnex 1440	. . . . . . 7 ⊢ (∀𝑧 ¬ ((𝑦 ∈ (2^nd ‘𝐴) ∧ (𝑦 +_Q 𝑞) ∈ (1^st ‘𝐵)) ∧ (𝑧 ∈ (1^st ‘𝐴) ∧ (𝑧 +_Q 𝑞) ∈ (2^nd ‘𝐵))) ↔ ¬ ∃𝑧((𝑦 ∈ (2^nd ‘𝐴) ∧ (𝑦 +_Q 𝑞) ∈ (1^st ‘𝐵)) ∧ (𝑧 ∈ (1^st ‘𝐴) ∧ (𝑧 +_Q 𝑞) ∈ (2^nd ‘𝐵))))
62	61	albii 1411	. . . . . 6 ⊢ (∀𝑦∀𝑧 ¬ ((𝑦 ∈ (2^nd ‘𝐴) ∧ (𝑦 +_Q 𝑞) ∈ (1^st ‘𝐵)) ∧ (𝑧 ∈ (1^st ‘𝐴) ∧ (𝑧 +_Q 𝑞) ∈ (2^nd ‘𝐵))) ↔ ∀𝑦 ¬ ∃𝑧((𝑦 ∈ (2^nd ‘𝐴) ∧ (𝑦 +_Q 𝑞) ∈ (1^st ‘𝐵)) ∧ (𝑧 ∈ (1^st ‘𝐴) ∧ (𝑧 +_Q 𝑞) ∈ (2^nd ‘𝐵))))
63		alnex 1440	. . . . . 6 ⊢ (∀𝑦 ¬ ∃𝑧((𝑦 ∈ (2^nd ‘𝐴) ∧ (𝑦 +_Q 𝑞) ∈ (1^st ‘𝐵)) ∧ (𝑧 ∈ (1^st ‘𝐴) ∧ (𝑧 +_Q 𝑞) ∈ (2^nd ‘𝐵))) ↔ ¬ ∃𝑦∃𝑧((𝑦 ∈ (2^nd ‘𝐴) ∧ (𝑦 +_Q 𝑞) ∈ (1^st ‘𝐵)) ∧ (𝑧 ∈ (1^st ‘𝐴) ∧ (𝑧 +_Q 𝑞) ∈ (2^nd ‘𝐵))))
64	62, 63	bitri 183	. . . . 5 ⊢ (∀𝑦∀𝑧 ¬ ((𝑦 ∈ (2^nd ‘𝐴) ∧ (𝑦 +_Q 𝑞) ∈ (1^st ‘𝐵)) ∧ (𝑧 ∈ (1^st ‘𝐴) ∧ (𝑧 +_Q 𝑞) ∈ (2^nd ‘𝐵))) ↔ ¬ ∃𝑦∃𝑧((𝑦 ∈ (2^nd ‘𝐴) ∧ (𝑦 +_Q 𝑞) ∈ (1^st ‘𝐵)) ∧ (𝑧 ∈ (1^st ‘𝐴) ∧ (𝑧 +_Q 𝑞) ∈ (2^nd ‘𝐵))))
65	60, 64	syl6bbr 197	. . . 4 ⊢ (𝑞 ∈ Q → (¬ (𝑞 ∈ (1^st ‘𝐶) ∧ 𝑞 ∈ (2^nd ‘𝐶)) ↔ ∀𝑦∀𝑧 ¬ ((𝑦 ∈ (2^nd ‘𝐴) ∧ (𝑦 +_Q 𝑞) ∈ (1^st ‘𝐵)) ∧ (𝑧 ∈ (1^st ‘𝐴) ∧ (𝑧 +_Q 𝑞) ∈ (2^nd ‘𝐵)))))
66	65	adantl 272	. . 3 ⊢ ((𝐴<_P 𝐵 ∧ 𝑞 ∈ Q) → (¬ (𝑞 ∈ (1^st ‘𝐶) ∧ 𝑞 ∈ (2^nd ‘𝐶)) ↔ ∀𝑦∀𝑧 ¬ ((𝑦 ∈ (2^nd ‘𝐴) ∧ (𝑦 +_Q 𝑞) ∈ (1^st ‘𝐵)) ∧ (𝑧 ∈ (1^st ‘𝐴) ∧ (𝑧 +_Q 𝑞) ∈ (2^nd ‘𝐵)))))
67	43, 66	mpbird 166	. 2 ⊢ ((𝐴<_P 𝐵 ∧ 𝑞 ∈ Q) → ¬ (𝑞 ∈ (1^st ‘𝐶) ∧ 𝑞 ∈ (2^nd ‘𝐶)))
68	67	ralrimiva 2458	1 ⊢ (𝐴<_P 𝐵 → ∀𝑞 ∈ Q ¬ (𝑞 ∈ (1^st ‘𝐶) ∧ 𝑞 ∈ (2^nd ‘𝐶)))

Colors of variables: wff set class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 103 ↔ wb 104 ∧ w3a 927 ∀wal 1294 = wceq 1296 ∃wex 1433 ∈ wcel 1445 ∀wral 2370 {crab 2374 ⟨cop 3469 class class class wbr 3867 ‘cfv 5049 (class class class)co 5690 1^st c1st 5947 2^nd c2nd 5948 Qcnq 6936 +_Q cplq 6938 <_Q cltq 6941 Pcnp 6947 <_P cltp 6951

This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 582 ax-in2 583 ax-io 668 ax-5 1388 ax-7 1389 ax-gen 1390 ax-ie1 1434 ax-ie2 1435 ax-8 1447 ax-10 1448 ax-11 1449 ax-i12 1450 ax-bndl 1451 ax-4 1452 ax-13 1456 ax-14 1457 ax-17 1471 ax-i9 1475 ax-ial 1479 ax-i5r 1480 ax-ext 2077 ax-coll 3975 ax-sep 3978 ax-nul 3986 ax-pow 4030 ax-pr 4060 ax-un 4284 ax-setind 4381 ax-iinf 4431

This theorem depends on definitions: df-bi 116 df-dc 784 df-3or 928 df-3an 929 df-tru 1299 df-fal 1302 df-nf 1402 df-sb 1700 df-eu 1958 df-mo 1959 df-clab 2082 df-cleq 2088 df-clel 2091 df-nfc 2224 df-ne 2263 df-ral 2375 df-rex 2376 df-reu 2377 df-rab 2379 df-v 2635 df-sbc 2855 df-csb 2948 df-dif 3015 df-un 3017 df-in 3019 df-ss 3026 df-nul 3303 df-pw 3451 df-sn 3472 df-pr 3473 df-op 3475 df-uni 3676 df-int 3711 df-iun 3754 df-br 3868 df-opab 3922 df-mpt 3923 df-tr 3959 df-eprel 4140 df-id 4144 df-po 4147 df-iso 4148 df-iord 4217 df-on 4219 df-suc 4222 df-iom 4434 df-xp 4473 df-rel 4474 df-cnv 4475 df-co 4476 df-dm 4477 df-rn 4478 df-res 4479 df-ima 4480 df-iota 5014 df-fun 5051 df-fn 5052 df-f 5053 df-f1 5054 df-fo 5055 df-f1o 5056 df-fv 5057 df-ov 5693 df-oprab 5694 df-mpt2 5695 df-1st 5949 df-2nd 5950 df-recs 6108 df-irdg 6173 df-oadd 6223 df-omul 6224 df-er 6332 df-ec 6334 df-qs 6338 df-ni 6960 df-pli 6961 df-mi 6962 df-lti 6963 df-plpq 7000 df-enq 7003 df-nqqs 7004 df-plqqs 7005 df-ltnqqs 7009 df-inp 7122 df-iltp 7126

This theorem is referenced by: ltexprlempr 7264

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