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Theorem ltexprlemdisj 7869

Description: Our constructed difference is disjoint. Lemma for ltexpri 7876. (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 7661	. . . . . 6 ⊢ <_Q Or Q
2		ltrelnq 7628	. . . . . 6 ⊢ <_Q ⊆ (Q × Q)
3	1, 2	son2lpi 5140	. . . . 5 ⊢ ¬ (𝑦 <_Q 𝑧 ∧ 𝑧 <_Q 𝑦)
4		ltrelpr 7768	. . . . . . . . . . . . . . . 16 ⊢ <_P ⊆ (P × P)
5	4	brel 4784	. . . . . . . . . . . . . . 15 ⊢ (𝐴<_P 𝐵 → (𝐴 ∈ P ∧ 𝐵 ∈ P))
6	5	simprd 114	. . . . . . . . . . . . . 14 ⊢ (𝐴<_P 𝐵 → 𝐵 ∈ P)
7		prop 7738	. . . . . . . . . . . . . 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 7750	. . . . . . . . . . . . 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 1307	. . . . . . . . . . . 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 1231	. . . . . . . . . . 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 7663	. . . . . . . . . 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 7738	. . . . . . . . . . . . 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 7745	. . . . . . . . . . . 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 7744	. . . . . . . . . . . 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 529	. . . . . . . . 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 7644	. . . . . . . . . 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 6202	. . . . . . . 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 7750	. . . . . . . . . . . . 13 ⊢ ((⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩ ∈ P ∧ 𝑧 ∈ (1^st ‘𝐴) ∧ 𝑦 ∈ (2^nd ‘𝐴)) → 𝑧 <_Q 𝑦)
34	19, 33	syl3an1 1307	. . . . . . . . . . . 12 ⊢ ((𝐴<_P 𝐵 ∧ 𝑧 ∈ (1^st ‘𝐴) ∧ 𝑦 ∈ (2^nd ‘𝐴)) → 𝑧 <_Q 𝑦)
35	34	3com23 1236	. . . . . . . . . . 11 ⊢ ((𝐴<_P 𝐵 ∧ 𝑦 ∈ (2^nd ‘𝐴) ∧ 𝑧 ∈ (1^st ‘𝐴)) → 𝑧 <_Q 𝑦)
36	35	3expb 1231	. . . . . . . . . 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 670	. . . 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 1923	. . 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 7861	. . . . . . . . . . 11 ⊢ (𝑞 ∈ (1^st ‘𝐶) ↔ (𝑞 ∈ Q ∧ ∃𝑦(𝑦 ∈ (2^nd ‘𝐴) ∧ (𝑦 +_Q 𝑞) ∈ (1^st ‘𝐵))))
46	44	ltexprlemelu 7862	. . . . . . . . . . 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 594	. . . . . . . . . 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 927	. . . . . . . 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 2294	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑧 → (𝑦 ∈ (1^st ‘𝐴) ↔ 𝑧 ∈ (1^st ‘𝐴)))
52		oveq1 6035	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑧 → (𝑦 +_Q 𝑞) = (𝑧 +_Q 𝑞))
53	52	eleq1d 2300	. . . . . . . . . . 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 1967	. . . . . . . . 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 1985	. . . . . . 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 673	. . . . 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 1548	. . . . . . 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 1519	. . . . . 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 1548	. . . . . 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 2606	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 1005 ∀wal 1396 = wceq 1398 ∃wex 1541 ∈ wcel 2202 ∀wral 2511 {crab 2515 ⟨cop 3676 class class class wbr 4093 ‘cfv 5333 (class class class)co 6028 1^st c1st 6310 2^nd c2nd 6311 Qcnq 7543 +_Q cplq 7545 <_Q cltq 7548 Pcnp 7554 <_P cltp 7558

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 619 ax-in2 620 ax-io 717 ax-5 1496 ax-7 1497 ax-gen 1498 ax-ie1 1542 ax-ie2 1543 ax-8 1553 ax-10 1554 ax-11 1555 ax-i12 1556 ax-bndl 1558 ax-4 1559 ax-17 1575 ax-i9 1579 ax-ial 1583 ax-i5r 1584 ax-13 2204 ax-14 2205 ax-ext 2213 ax-coll 4209 ax-sep 4212 ax-nul 4220 ax-pow 4270 ax-pr 4305 ax-un 4536 ax-setind 4641 ax-iinf 4692

This theorem depends on definitions: df-bi 117 df-dc 843 df-3or 1006 df-3an 1007 df-tru 1401 df-fal 1404 df-nf 1510 df-sb 1811 df-eu 2082 df-mo 2083 df-clab 2218 df-cleq 2224 df-clel 2227 df-nfc 2364 df-ne 2404 df-ral 2516 df-rex 2517 df-reu 2518 df-rab 2520 df-v 2805 df-sbc 3033 df-csb 3129 df-dif 3203 df-un 3205 df-in 3207 df-ss 3214 df-nul 3497 df-pw 3658 df-sn 3679 df-pr 3680 df-op 3682 df-uni 3899 df-int 3934 df-iun 3977 df-br 4094 df-opab 4156 df-mpt 4157 df-tr 4193 df-eprel 4392 df-id 4396 df-po 4399 df-iso 4400 df-iord 4469 df-on 4471 df-suc 4474 df-iom 4695 df-xp 4737 df-rel 4738 df-cnv 4739 df-co 4740 df-dm 4741 df-rn 4742 df-res 4743 df-ima 4744 df-iota 5293 df-fun 5335 df-fn 5336 df-f 5337 df-f1 5338 df-fo 5339 df-f1o 5340 df-fv 5341 df-ov 6031 df-oprab 6032 df-mpo 6033 df-1st 6312 df-2nd 6313 df-recs 6514 df-irdg 6579 df-oadd 6629 df-omul 6630 df-er 6745 df-ec 6747 df-qs 6751 df-ni 7567 df-pli 7568 df-mi 7569 df-lti 7570 df-plpq 7607 df-enq 7610 df-nqqs 7611 df-plqqs 7612 df-ltnqqs 7616 df-inp 7729 df-iltp 7733

This theorem is referenced by: ltexprlempr 7871

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