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Theorem ltexprlemdisj 7438

Description: Our constructed difference is disjoint. Lemma for ltexpri 7445. (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 7230	. . . . . 6 ⊢ <_Q Or Q
2		ltrelnq 7197	. . . . . 6 ⊢ <_Q ⊆ (Q × Q)
3	1, 2	son2lpi 4943	. . . . 5 ⊢ ¬ (𝑦 <_Q 𝑧 ∧ 𝑧 <_Q 𝑦)
4		ltrelpr 7337	. . . . . . . . . . . . . . . 16 ⊢ <_P ⊆ (P × P)
5	4	brel 4599	. . . . . . . . . . . . . . 15 ⊢ (𝐴<_P 𝐵 → (𝐴 ∈ P ∧ 𝐵 ∈ P))
6	5	simprd 113	. . . . . . . . . . . . . 14 ⊢ (𝐴<_P 𝐵 → 𝐵 ∈ P)
7		prop 7307	. . . . . . . . . . . . . 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 7319	. . . . . . . . . . . . 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 1250	. . . . . . . . . . . 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 1183	. . . . . . . . . . 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 469	. . . . . . . . . 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 476	. . . . . . . . 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 478	. . . . . . . 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 7232	. . . . . . . . . 10 ⊢ ((𝑓 ∈ Q ∧ 𝑔 ∈ Q ∧ ℎ ∈ Q) → (𝑓 <_Q 𝑔 ↔ (ℎ +_Q 𝑓) <_Q (ℎ +_Q 𝑔)))
16	15	adantl 275	. . . . . . . . 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 7307	. . . . . . . . . . . . 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 7314	. . . . . . . . . . . 12 ⊢ ((⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩ ∈ P ∧ 𝑦 ∈ (2^nd ‘𝐴)) → 𝑦 ∈ Q)
21	19, 20	sylan 281	. . . . . . . . . . 11 ⊢ ((𝐴<_P 𝐵 ∧ 𝑦 ∈ (2^nd ‘𝐴)) → 𝑦 ∈ Q)
22	21	ad2ant2r 501	. . . . . . . . . 10 ⊢ (((𝐴<_P 𝐵 ∧ 𝑞 ∈ Q) ∧ (𝑦 ∈ (2^nd ‘𝐴) ∧ (𝑦 +_Q 𝑞) ∈ (1^st ‘𝐵))) → 𝑦 ∈ Q)
23	22	adantrr 471	. . . . . . . . 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 7313	. . . . . . . . . . . 12 ⊢ ((⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩ ∈ P ∧ 𝑧 ∈ (1^st ‘𝐴)) → 𝑧 ∈ Q)
25	19, 24	sylan 281	. . . . . . . . . . 11 ⊢ ((𝐴<_P 𝐵 ∧ 𝑧 ∈ (1^st ‘𝐴)) → 𝑧 ∈ Q)
26	25	ad2ant2r 501	. . . . . . . . . 10 ⊢ (((𝐴<_P 𝐵 ∧ 𝑞 ∈ Q) ∧ (𝑧 ∈ (1^st ‘𝐴) ∧ (𝑧 +_Q 𝑞) ∈ (2^nd ‘𝐵))) → 𝑧 ∈ Q)
27	26	adantrl 470	. . . . . . . . 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 520	. . . . . . . . 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 7213	. . . . . . . . . 10 ⊢ ((𝑓 ∈ Q ∧ 𝑔 ∈ Q) → (𝑓 +_Q 𝑔) = (𝑔 +_Q 𝑓))
30	29	adantl 275	. . . . . . . . 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 5948	. . . . . . . 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 7319	. . . . . . . . . . . . 13 ⊢ ((⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩ ∈ P ∧ 𝑧 ∈ (1^st ‘𝐴) ∧ 𝑦 ∈ (2^nd ‘𝐴)) → 𝑧 <_Q 𝑦)
34	19, 33	syl3an1 1250	. . . . . . . . . . . 12 ⊢ ((𝐴<_P 𝐵 ∧ 𝑧 ∈ (1^st ‘𝐴) ∧ 𝑦 ∈ (2^nd ‘𝐴)) → 𝑧 <_Q 𝑦)
35	34	3com23 1188	. . . . . . . . . . 11 ⊢ ((𝐴<_P 𝐵 ∧ 𝑦 ∈ (2^nd ‘𝐴) ∧ 𝑧 ∈ (1^st ‘𝐴)) → 𝑧 <_Q 𝑦)
36	35	3expb 1183	. . . . . . . . . 10 ⊢ ((𝐴<_P 𝐵 ∧ (𝑦 ∈ (2^nd ‘𝐴) ∧ 𝑧 ∈ (1^st ‘𝐴))) → 𝑧 <_Q 𝑦)
37	36	adantlr 469	. . . . . . . . 9 ⊢ (((𝐴<_P 𝐵 ∧ 𝑞 ∈ Q) ∧ (𝑦 ∈ (2^nd ‘𝐴) ∧ 𝑧 ∈ (1^st ‘𝐴))) → 𝑧 <_Q 𝑦)
38	37	adantrlr 477	. . . . . . . 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 479	. . . . . . 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 304	. . . . . 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 654	. . . 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 1848	. . 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 7430	. . . . . . . . . . 11 ⊢ (𝑞 ∈ (1^st ‘𝐶) ↔ (𝑞 ∈ Q ∧ ∃𝑦(𝑦 ∈ (2^nd ‘𝐴) ∧ (𝑦 +_Q 𝑞) ∈ (1^st ‘𝐵))))
46	44	ltexprlemelu 7431	. . . . . . . . . . 11 ⊢ (𝑞 ∈ (2^nd ‘𝐶) ↔ (𝑞 ∈ Q ∧ ∃𝑦(𝑦 ∈ (1^st ‘𝐴) ∧ (𝑦 +_Q 𝑞) ∈ (2^nd ‘𝐵))))
47	45, 46	anbi12i 456	. . . . . . . . . 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 580	. . . . . . . . . 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 905	. . . . . . . 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 2203	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑧 → (𝑦 ∈ (1^st ‘𝐴) ↔ 𝑧 ∈ (1^st ‘𝐴)))
52		oveq1 5789	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑧 → (𝑦 +_Q 𝑞) = (𝑧 +_Q 𝑞))
53	52	eleq1d 2209	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑧 → ((𝑦 +_Q 𝑞) ∈ (2^nd ‘𝐵) ↔ (𝑧 +_Q 𝑞) ∈ (2^nd ‘𝐵)))
54	51, 53	anbi12d 465	. . . . . . . . . 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 1891	. . . . . . . . 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 453	. . . . . . . 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 1905	. . . . . . 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 657	. . . . 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 1476	. . . . . . 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 1447	. . . . . 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 1476	. . . . . 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 275	. . 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 2508	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 963 ∀wal 1330 = wceq 1332 ∃wex 1469 ∈ wcel 1481 ∀wral 2417 {crab 2421 ⟨cop 3535 class class class wbr 3937 ‘cfv 5131 (class class class)co 5782 1^st c1st 6044 2^nd c2nd 6045 Qcnq 7112 +_Q cplq 7114 <_Q cltq 7117 Pcnp 7123 <_P cltp 7127

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 604 ax-in2 605 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-13 1492 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-coll 4051 ax-sep 4054 ax-nul 4062 ax-pow 4106 ax-pr 4139 ax-un 4363 ax-setind 4460 ax-iinf 4510

This theorem depends on definitions: df-bi 116 df-dc 821 df-3or 964 df-3an 965 df-tru 1335 df-fal 1338 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ne 2310 df-ral 2422 df-rex 2423 df-reu 2424 df-rab 2426 df-v 2691 df-sbc 2914 df-csb 3008 df-dif 3078 df-un 3080 df-in 3082 df-ss 3089 df-nul 3369 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-int 3780 df-iun 3823 df-br 3938 df-opab 3998 df-mpt 3999 df-tr 4035 df-eprel 4219 df-id 4223 df-po 4226 df-iso 4227 df-iord 4296 df-on 4298 df-suc 4301 df-iom 4513 df-xp 4553 df-rel 4554 df-cnv 4555 df-co 4556 df-dm 4557 df-rn 4558 df-res 4559 df-ima 4560 df-iota 5096 df-fun 5133 df-fn 5134 df-f 5135 df-f1 5136 df-fo 5137 df-f1o 5138 df-fv 5139 df-ov 5785 df-oprab 5786 df-mpo 5787 df-1st 6046 df-2nd 6047 df-recs 6210 df-irdg 6275 df-oadd 6325 df-omul 6326 df-er 6437 df-ec 6439 df-qs 6443 df-ni 7136 df-pli 7137 df-mi 7138 df-lti 7139 df-plpq 7176 df-enq 7179 df-nqqs 7180 df-plqqs 7181 df-ltnqqs 7185 df-inp 7298 df-iltp 7302

This theorem is referenced by: ltexprlempr 7440

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