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Theorem ltexprlemdisj 7816

Description: Our constructed difference is disjoint. Lemma for ltexpri 7823. (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 7608	. . . . . 6 ⊢ <_Q Or Q
2		ltrelnq 7575	. . . . . 6 ⊢ <_Q ⊆ (Q × Q)
3	1, 2	son2lpi 5131	. . . . 5 ⊢ ¬ (𝑦 <_Q 𝑧 ∧ 𝑧 <_Q 𝑦)
4		ltrelpr 7715	. . . . . . . . . . . . . . . 16 ⊢ <_P ⊆ (P × P)
5	4	brel 4776	. . . . . . . . . . . . . . 15 ⊢ (𝐴<_P 𝐵 → (𝐴 ∈ P ∧ 𝐵 ∈ P))
6	5	simprd 114	. . . . . . . . . . . . . 14 ⊢ (𝐴<_P 𝐵 → 𝐵 ∈ P)
7		prop 7685	. . . . . . . . . . . . . 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 7697	. . . . . . . . . . . . 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 1304	. . . . . . . . . . . 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 1228	. . . . . . . . . . 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 477	. . . . . . . . . 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 484	. . . . . . . . 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 486	. . . . . . . 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 7610	. . . . . . . . . 10 ⊢ ((𝑓 ∈ Q ∧ 𝑔 ∈ Q ∧ ℎ ∈ Q) → (𝑓 <_Q 𝑔 ↔ (ℎ +_Q 𝑓) <_Q (ℎ +_Q 𝑔)))
16	15	adantl 277	. . . . . . . . 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 112	. . . . . . . . . . . . 13 ⊢ (𝐴<_P 𝐵 → 𝐴 ∈ P)
18		prop 7685	. . . . . . . . . . . . 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 7692	. . . . . . . . . . . 12 ⊢ ((⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩ ∈ P ∧ 𝑦 ∈ (2^nd ‘𝐴)) → 𝑦 ∈ Q)
21	19, 20	sylan 283	. . . . . . . . . . 11 ⊢ ((𝐴<_P 𝐵 ∧ 𝑦 ∈ (2^nd ‘𝐴)) → 𝑦 ∈ Q)
22	21	ad2ant2r 509	. . . . . . . . . 10 ⊢ (((𝐴<_P 𝐵 ∧ 𝑞 ∈ Q) ∧ (𝑦 ∈ (2^nd ‘𝐴) ∧ (𝑦 +_Q 𝑞) ∈ (1^st ‘𝐵))) → 𝑦 ∈ Q)
23	22	adantrr 479	. . . . . . . . 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 7691	. . . . . . . . . . . 12 ⊢ ((⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩ ∈ P ∧ 𝑧 ∈ (1^st ‘𝐴)) → 𝑧 ∈ Q)
25	19, 24	sylan 283	. . . . . . . . . . 11 ⊢ ((𝐴<_P 𝐵 ∧ 𝑧 ∈ (1^st ‘𝐴)) → 𝑧 ∈ Q)
26	25	ad2ant2r 509	. . . . . . . . . 10 ⊢ (((𝐴<_P 𝐵 ∧ 𝑞 ∈ Q) ∧ (𝑧 ∈ (1^st ‘𝐴) ∧ (𝑧 +_Q 𝑞) ∈ (2^nd ‘𝐵))) → 𝑧 ∈ Q)
27	26	adantrl 478	. . . . . . . . 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 528	. . . . . . . . 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 7591	. . . . . . . . . 10 ⊢ ((𝑓 ∈ Q ∧ 𝑔 ∈ Q) → (𝑓 +_Q 𝑔) = (𝑔 +_Q 𝑓))
30	29	adantl 277	. . . . . . . . 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 6187	. . . . . . . 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 167	. . . . . . 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 7697	. . . . . . . . . . . . 13 ⊢ ((⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩ ∈ P ∧ 𝑧 ∈ (1^st ‘𝐴) ∧ 𝑦 ∈ (2^nd ‘𝐴)) → 𝑧 <_Q 𝑦)
34	19, 33	syl3an1 1304	. . . . . . . . . . . 12 ⊢ ((𝐴<_P 𝐵 ∧ 𝑧 ∈ (1^st ‘𝐴) ∧ 𝑦 ∈ (2^nd ‘𝐴)) → 𝑧 <_Q 𝑦)
35	34	3com23 1233	. . . . . . . . . . 11 ⊢ ((𝐴<_P 𝐵 ∧ 𝑦 ∈ (2^nd ‘𝐴) ∧ 𝑧 ∈ (1^st ‘𝐴)) → 𝑧 <_Q 𝑦)
36	35	3expb 1228	. . . . . . . . . 10 ⊢ ((𝐴<_P 𝐵 ∧ (𝑦 ∈ (2^nd ‘𝐴) ∧ 𝑧 ∈ (1^st ‘𝐴))) → 𝑧 <_Q 𝑦)
37	36	adantlr 477	. . . . . . . . 9 ⊢ (((𝐴<_P 𝐵 ∧ 𝑞 ∈ Q) ∧ (𝑦 ∈ (2^nd ‘𝐴) ∧ 𝑧 ∈ (1^st ‘𝐴))) → 𝑧 <_Q 𝑦)
38	37	adantrlr 485	. . . . . . . 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 487	. . . . . . 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 306	. . . . . 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 115	. . . . 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 668	. . . 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 1921	. . 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 7808	. . . . . . . . . . 11 ⊢ (𝑞 ∈ (1^st ‘𝐶) ↔ (𝑞 ∈ Q ∧ ∃𝑦(𝑦 ∈ (2^nd ‘𝐴) ∧ (𝑦 +_Q 𝑞) ∈ (1^st ‘𝐵))))
46	44	ltexprlemelu 7809	. . . . . . . . . . 11 ⊢ (𝑞 ∈ (2^nd ‘𝐶) ↔ (𝑞 ∈ Q ∧ ∃𝑦(𝑦 ∈ (1^st ‘𝐴) ∧ (𝑦 +_Q 𝑞) ∈ (2^nd ‘𝐵))))
47	45, 46	anbi12i 460	. . . . . . . . . 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 592	. . . . . . . . . 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 187	. . . . . . . . 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 924	. . . . . . . 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 2292	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑧 → (𝑦 ∈ (1^st ‘𝐴) ↔ 𝑧 ∈ (1^st ‘𝐴)))
52		oveq1 6020	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑧 → (𝑦 +_Q 𝑞) = (𝑧 +_Q 𝑞))
53	52	eleq1d 2298	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑧 → ((𝑦 +_Q 𝑞) ∈ (2^nd ‘𝐵) ↔ (𝑧 +_Q 𝑞) ∈ (2^nd ‘𝐵)))
54	51, 53	anbi12d 473	. . . . . . . . . 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 1965	. . . . . . . . 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 457	. . . . . . . 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	bitrdi 196	. . . . . . 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 1983	. . . . . . 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	bitr4di 198	. . . . . 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 671	. . . . 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 1545	. . . . . . 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 1516	. . . . . 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 1545	. . . . . 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 184	. . . . 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	bitr4di 198	. . . 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 277	. . 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 167	. 2 ⊢ ((𝐴<_P 𝐵 ∧ 𝑞 ∈ Q) → ¬ (𝑞 ∈ (1^st ‘𝐶) ∧ 𝑞 ∈ (2^nd ‘𝐶)))
68	67	ralrimiva 2603	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 104 ↔ wb 105 ∧ w3a 1002 ∀wal 1393 = wceq 1395 ∃wex 1538 ∈ wcel 2200 ∀wral 2508 {crab 2512 ⟨cop 3670 class class class wbr 4086 ‘cfv 5324 (class class class)co 6013 1^st c1st 6296 2^nd c2nd 6297 Qcnq 7490 +_Q cplq 7492 <_Q cltq 7495 Pcnp 7501 <_P cltp 7505

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 617 ax-in2 618 ax-io 714 ax-5 1493 ax-7 1494 ax-gen 1495 ax-ie1 1539 ax-ie2 1540 ax-8 1550 ax-10 1551 ax-11 1552 ax-i12 1553 ax-bndl 1555 ax-4 1556 ax-17 1572 ax-i9 1576 ax-ial 1580 ax-i5r 1581 ax-13 2202 ax-14 2203 ax-ext 2211 ax-coll 4202 ax-sep 4205 ax-nul 4213 ax-pow 4262 ax-pr 4297 ax-un 4528 ax-setind 4633 ax-iinf 4684

This theorem depends on definitions: df-bi 117 df-dc 840 df-3or 1003 df-3an 1004 df-tru 1398 df-fal 1401 df-nf 1507 df-sb 1809 df-eu 2080 df-mo 2081 df-clab 2216 df-cleq 2222 df-clel 2225 df-nfc 2361 df-ne 2401 df-ral 2513 df-rex 2514 df-reu 2515 df-rab 2517 df-v 2802 df-sbc 3030 df-csb 3126 df-dif 3200 df-un 3202 df-in 3204 df-ss 3211 df-nul 3493 df-pw 3652 df-sn 3673 df-pr 3674 df-op 3676 df-uni 3892 df-int 3927 df-iun 3970 df-br 4087 df-opab 4149 df-mpt 4150 df-tr 4186 df-eprel 4384 df-id 4388 df-po 4391 df-iso 4392 df-iord 4461 df-on 4463 df-suc 4466 df-iom 4687 df-xp 4729 df-rel 4730 df-cnv 4731 df-co 4732 df-dm 4733 df-rn 4734 df-res 4735 df-ima 4736 df-iota 5284 df-fun 5326 df-fn 5327 df-f 5328 df-f1 5329 df-fo 5330 df-f1o 5331 df-fv 5332 df-ov 6016 df-oprab 6017 df-mpo 6018 df-1st 6298 df-2nd 6299 df-recs 6466 df-irdg 6531 df-oadd 6581 df-omul 6582 df-er 6697 df-ec 6699 df-qs 6703 df-ni 7514 df-pli 7515 df-mi 7516 df-lti 7517 df-plpq 7554 df-enq 7557 df-nqqs 7558 df-plqqs 7559 df-ltnqqs 7563 df-inp 7676 df-iltp 7680

This theorem is referenced by: ltexprlempr 7818

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