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Theorem ltexprlemdisj 6578

Description: Our constructed difference is disjoint. Lemma for ltexpri 6585. (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 6382	. . . . . 6 ⊢ <_Q Or Q
2		ltrelnq 6349	. . . . . 6 ⊢ <_Q ⊆ (Q × Q)
3	1, 2	son2lpi 4664	. . . . 5 ⊢ ¬ (y <_Q z ∧ z <_Q y)
4		ltrelpr 6487	. . . . . . . . . . . . . . . 16 ⊢ <_P ⊆ (P × P)
5	4	brel 4335	. . . . . . . . . . . . . . 15 ⊢ (A<_P B → (A ∈ P ∧ B ∈ P))
6	5	simprd 107	. . . . . . . . . . . . . 14 ⊢ (A<_P B → B ∈ P)
7		prop 6457	. . . . . . . . . . . . . 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 6469	. . . . . . . . . . . . 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 1167	. . . . . . . . . . . 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 1104	. . . . . . . . . . 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 446	. . . . . . . . . 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 453	. . . . . . . . 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 455	. . . . . . . 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 6384	. . . . . . . . . 10 ⊢ ((f ∈ Q ∧ g ∈ Q ∧ ℎ ∈ Q) → (f <_Q g ↔ (ℎ +_Q f) <_Q (ℎ +_Q g)))
16	15	adantl 262	. . . . . . . . 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 105	. . . . . . . . . . . . 13 ⊢ (A<_P B → A ∈ P)
18		prop 6457	. . . . . . . . . . . . 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 6464	. . . . . . . . . . . 12 ⊢ ((⟨(1^st ‘A), (2^nd ‘A)⟩ ∈ P ∧ y ∈ (2^nd ‘A)) → y ∈ Q)
21	19, 20	sylan 267	. . . . . . . . . . 11 ⊢ ((A<_P B ∧ y ∈ (2^nd ‘A)) → y ∈ Q)
22	21	ad2ant2r 478	. . . . . . . . . 10 ⊢ (((A<_P B ∧ 𝑞 ∈ Q) ∧ (y ∈ (2^nd ‘A) ∧ (y +_Q 𝑞) ∈ (1^st ‘B))) → y ∈ Q)
23	22	adantrr 448	. . . . . . . . 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 6463	. . . . . . . . . . . 12 ⊢ ((⟨(1^st ‘A), (2^nd ‘A)⟩ ∈ P ∧ z ∈ (1^st ‘A)) → z ∈ Q)
25	19, 24	sylan 267	. . . . . . . . . . 11 ⊢ ((A<_P B ∧ z ∈ (1^st ‘A)) → z ∈ Q)
26	25	ad2ant2r 478	. . . . . . . . . 10 ⊢ (((A<_P B ∧ 𝑞 ∈ Q) ∧ (z ∈ (1^st ‘A) ∧ (z +_Q 𝑞) ∈ (2^nd ‘B))) → z ∈ Q)
27	26	adantrl 447	. . . . . . . . 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 482	. . . . . . . . 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 6365	. . . . . . . . . 10 ⊢ ((f ∈ Q ∧ g ∈ Q) → (f +_Q g) = (g +_Q f))
30	29	adantl 262	. . . . . . . . 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 5612	. . . . . . . 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 156	. . . . . . 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 6469	. . . . . . . . . . . . 13 ⊢ ((⟨(1^st ‘A), (2^nd ‘A)⟩ ∈ P ∧ z ∈ (1^st ‘A) ∧ y ∈ (2^nd ‘A)) → z <_Q y)
34	19, 33	syl3an1 1167	. . . . . . . . . . . 12 ⊢ ((A<_P B ∧ z ∈ (1^st ‘A) ∧ y ∈ (2^nd ‘A)) → z <_Q y)
35	34	3com23 1109	. . . . . . . . . . 11 ⊢ ((A<_P B ∧ y ∈ (2^nd ‘A) ∧ z ∈ (1^st ‘A)) → z <_Q y)
36	35	3expb 1104	. . . . . . . . . 10 ⊢ ((A<_P B ∧ (y ∈ (2^nd ‘A) ∧ z ∈ (1^st ‘A))) → z <_Q y)
37	36	adantlr 446	. . . . . . . . 9 ⊢ (((A<_P B ∧ 𝑞 ∈ Q) ∧ (y ∈ (2^nd ‘A) ∧ z ∈ (1^st ‘A))) → z <_Q y)
38	37	adantrlr 454	. . . . . . . 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 456	. . . . . . 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 290	. . . . . 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 108	. . . . 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 589	. . . 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 1752	. . 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 6570	. . . . . . . . . . 11 ⊢ (𝑞 ∈ (1^st ‘𝐶) ↔ (𝑞 ∈ Q ∧ ∃y(y ∈ (2^nd ‘A) ∧ (y +_Q 𝑞) ∈ (1^st ‘B))))
46	44	ltexprlemelu 6571	. . . . . . . . . . 11 ⊢ (𝑞 ∈ (2^nd ‘𝐶) ↔ (𝑞 ∈ Q ∧ ∃y(y ∈ (1^st ‘A) ∧ (y +_Q 𝑞) ∈ (2^nd ‘B))))
47	45, 46	anbi12i 433	. . . . . . . . . 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 524	. . . . . . . . . 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 176	. . . . . . . . 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 827	. . . . . . . 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 2097	. . . . . . . . . . 11 ⊢ (y = z → (y ∈ (1^st ‘A) ↔ z ∈ (1^st ‘A)))
52		oveq1 5462	. . . . . . . . . . . 12 ⊢ (y = z → (y +_Q 𝑞) = (z +_Q 𝑞))
53	52	eleq1d 2103	. . . . . . . . . . 11 ⊢ (y = z → ((y +_Q 𝑞) ∈ (2^nd ‘B) ↔ (z +_Q 𝑞) ∈ (2^nd ‘B)))
54	51, 53	anbi12d 442	. . . . . . . . . 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 1792	. . . . . . . . 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 430	. . . . . . . 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 185	. . . . . . 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 1804	. . . . . . 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 187	. . . . . 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 591	. . . . 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 1385	. . . . . . 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 1356	. . . . . 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 1385	. . . . . 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 173	. . . . 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 187	. . . 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 262	. . 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 156	. 2 ⊢ ((A<_P B ∧ 𝑞 ∈ Q) → ¬ (𝑞 ∈ (1^st ‘𝐶) ∧ 𝑞 ∈ (2^nd ‘𝐶)))
68	67	ralrimiva 2386	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 97 ↔ wb 98 ∧ w3a 884 ∀wal 1240 = wceq 1242 ∃wex 1378 ∈ wcel 1390 ∀wral 2300 {crab 2304 ⟨cop 3370 class class class wbr 3755 ‘cfv 4845 (class class class)co 5455 1^st c1st 5707 2^nd c2nd 5708 Qcnq 6264 +_Q cplq 6266 <_Q cltq 6269 Pcnp 6275 <_P cltp 6279

This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bnd 1396 ax-4 1397 ax-13 1401 ax-14 1402 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 ax-coll 3863 ax-sep 3866 ax-nul 3874 ax-pow 3918 ax-pr 3935 ax-un 4136 ax-setind 4220 ax-iinf 4254

This theorem depends on definitions: df-bi 110 df-dc 742 df-3or 885 df-3an 886 df-tru 1245 df-fal 1248 df-nf 1347 df-sb 1643 df-eu 1900 df-mo 1901 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-ne 2203 df-ral 2305 df-rex 2306 df-reu 2307 df-rab 2309 df-v 2553 df-sbc 2759 df-csb 2847 df-dif 2914 df-un 2916 df-in 2918 df-ss 2925 df-nul 3219 df-pw 3353 df-sn 3373 df-pr 3374 df-op 3376 df-uni 3572 df-int 3607 df-iun 3650 df-br 3756 df-opab 3810 df-mpt 3811 df-tr 3846 df-eprel 4017 df-id 4021 df-po 4024 df-iso 4025 df-iord 4069 df-on 4071 df-suc 4074 df-iom 4257 df-xp 4294 df-rel 4295 df-cnv 4296 df-co 4297 df-dm 4298 df-rn 4299 df-res 4300 df-ima 4301 df-iota 4810 df-fun 4847 df-fn 4848 df-f 4849 df-f1 4850 df-fo 4851 df-f1o 4852 df-fv 4853 df-ov 5458 df-oprab 5459 df-mpt2 5460 df-1st 5709 df-2nd 5710 df-recs 5861 df-irdg 5897 df-oadd 5944 df-omul 5945 df-er 6042 df-ec 6044 df-qs 6048 df-ni 6288 df-pli 6289 df-mi 6290 df-lti 6291 df-plpq 6328 df-enq 6331 df-nqqs 6332 df-plqqs 6333 df-ltnqqs 6337 df-inp 6448 df-iltp 6452

This theorem is referenced by: ltexprlempr 6580

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