Theorem ltsonq 11081

Description: 'Less than' is a strict ordering on positive fractions. (Contributed by NM, 19-Feb-1996.) (Revised by Mario Carneiro, 4-May-2013.) (New usage is discouraged.)

Assertion

Ref	Expression
ltsonq	⊢ <Q Or Q

Proof of Theorem ltsonq

Dummy variables 𝑠 𝑟 𝑡 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elpqn 11037	. . . . . . 7 ⊢ (𝑥 ∈ Q → 𝑥 ∈ (N × N))
2	1	adantr 486	. . . . . 6 ⊢ ((𝑥 ∈ Q ∧ 𝑦 ∈ Q) → 𝑥 ∈ (N × N))
3		xp1st 8057	. . . . . 6 ⊢ (𝑥 ∈ (N × N) → (1st ‘𝑥) ∈ N)
4	2, 3	syl 18	. . . . 5 ⊢ ((𝑥 ∈ Q ∧ 𝑦 ∈ Q) → (1st ‘𝑥) ∈ N)
5		elpqn 11037	. . . . . . 7 ⊢ (𝑦 ∈ Q → 𝑦 ∈ (N × N))
6	5	adantl 487	. . . . . 6 ⊢ ((𝑥 ∈ Q ∧ 𝑦 ∈ Q) → 𝑦 ∈ (N × N))
7		xp2nd 8058	. . . . . 6 ⊢ (𝑦 ∈ (N × N) → (2nd ‘𝑦) ∈ N)
8	6, 7	syl 18	. . . . 5 ⊢ ((𝑥 ∈ Q ∧ 𝑦 ∈ Q) → (2nd ‘𝑦) ∈ N)
9		mulclpi 11005	. . . . 5 ⊢ (((1st ‘𝑥) ∈ N ∧ (2nd ‘𝑦) ∈ N) → ((1st ‘𝑥) ·N (2nd ‘𝑦)) ∈ N)
10	4, 8, 9	syl2anc 596	. . . 4 ⊢ ((𝑥 ∈ Q ∧ 𝑦 ∈ Q) → ((1st ‘𝑥) ·N (2nd ‘𝑦)) ∈ N)
11		xp1st 8057	. . . . . 6 ⊢ (𝑦 ∈ (N × N) → (1st ‘𝑦) ∈ N)
12	6, 11	syl 18	. . . . 5 ⊢ ((𝑥 ∈ Q ∧ 𝑦 ∈ Q) → (1st ‘𝑦) ∈ N)
13		xp2nd 8058	. . . . . 6 ⊢ (𝑥 ∈ (N × N) → (2nd ‘𝑥) ∈ N)
14	2, 13	syl 18	. . . . 5 ⊢ ((𝑥 ∈ Q ∧ 𝑦 ∈ Q) → (2nd ‘𝑥) ∈ N)
15		mulclpi 11005	. . . . 5 ⊢ (((1st ‘𝑦) ∈ N ∧ (2nd ‘𝑥) ∈ N) → ((1st ‘𝑦) ·N (2nd ‘𝑥)) ∈ N)
16	12, 14, 15	syl2anc 596	. . . 4 ⊢ ((𝑥 ∈ Q ∧ 𝑦 ∈ Q) → ((1st ‘𝑦) ·N (2nd ‘𝑥)) ∈ N)
17		ltsopi 11000	. . . . 5 ⊢ <N Or N
18		sotric 5593	. . . . 5 ⊢ (( <N Or N ∧ (((1st ‘𝑥) ·N (2nd ‘𝑦)) ∈ N ∧ ((1st ‘𝑦) ·N (2nd ‘𝑥)) ∈ N)) → (((1st ‘𝑥) ·N (2nd ‘𝑦)) <N ((1st ‘𝑦) ·N (2nd ‘𝑥)) ↔ ¬ (((1st ‘𝑥) ·N (2nd ‘𝑦)) = ((1st ‘𝑦) ·N (2nd ‘𝑥)) ∨ ((1st ‘𝑦) ·N (2nd ‘𝑥)) <N ((1st ‘𝑥) ·N (2nd ‘𝑦)))))
19	17, 18	mpan 703	. . . 4 ⊢ ((((1st ‘𝑥) ·N (2nd ‘𝑦)) ∈ N ∧ ((1st ‘𝑦) ·N (2nd ‘𝑥)) ∈ N) → (((1st ‘𝑥) ·N (2nd ‘𝑦)) <N ((1st ‘𝑦) ·N (2nd ‘𝑥)) ↔ ¬ (((1st ‘𝑥) ·N (2nd ‘𝑦)) = ((1st ‘𝑦) ·N (2nd ‘𝑥)) ∨ ((1st ‘𝑦) ·N (2nd ‘𝑥)) <N ((1st ‘𝑥) ·N (2nd ‘𝑦)))))
20	10, 16, 19	syl2anc 596	. . 3 ⊢ ((𝑥 ∈ Q ∧ 𝑦 ∈ Q) → (((1st ‘𝑥) ·N (2nd ‘𝑦)) <N ((1st ‘𝑦) ·N (2nd ‘𝑥)) ↔ ¬ (((1st ‘𝑥) ·N (2nd ‘𝑦)) = ((1st ‘𝑦) ·N (2nd ‘𝑥)) ∨ ((1st ‘𝑦) ·N (2nd ‘𝑥)) <N ((1st ‘𝑥) ·N (2nd ‘𝑦)))))
21		ordpinq 11055	. . 3 ⊢ ((𝑥 ∈ Q ∧ 𝑦 ∈ Q) → (𝑥 <Q 𝑦 ↔ ((1st ‘𝑥) ·N (2nd ‘𝑦)) <N ((1st ‘𝑦) ·N (2nd ‘𝑥))))
22		fveq2 6908	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (1st ‘𝑥) = (1st ‘𝑦))
23		fveq2 6908	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (2nd ‘𝑥) = (2nd ‘𝑦))
24	23	eqcomd 2769	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (2nd ‘𝑦) = (2nd ‘𝑥))
25	22, 24	oveq12d 7461	. . . . . 6 ⊢ (𝑥 = 𝑦 → ((1st ‘𝑥) ·N (2nd ‘𝑦)) = ((1st ‘𝑦) ·N (2nd ‘𝑥)))
26		enqbreq2 11032	. . . . . . . 8 ⊢ ((𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)) → (𝑥 ~Q 𝑦 ↔ ((1st ‘𝑥) ·N (2nd ‘𝑦)) = ((1st ‘𝑦) ·N (2nd ‘𝑥))))
27	1, 5, 26	syl2an 608	. . . . . . 7 ⊢ ((𝑥 ∈ Q ∧ 𝑦 ∈ Q) → (𝑥 ~Q 𝑦 ↔ ((1st ‘𝑥) ·N (2nd ‘𝑦)) = ((1st ‘𝑦) ·N (2nd ‘𝑥))))
28		enqeq 11046	. . . . . . . 8 ⊢ ((𝑥 ∈ Q ∧ 𝑦 ∈ Q ∧ 𝑥 ~Q 𝑦) → 𝑥 = 𝑦)
29	28	3expia 1139	. . . . . . 7 ⊢ ((𝑥 ∈ Q ∧ 𝑦 ∈ Q) → (𝑥 ~Q 𝑦 → 𝑥 = 𝑦))
30	27, 29	sylbird 263	. . . . . 6 ⊢ ((𝑥 ∈ Q ∧ 𝑦 ∈ Q) → (((1st ‘𝑥) ·N (2nd ‘𝑦)) = ((1st ‘𝑦) ·N (2nd ‘𝑥)) → 𝑥 = 𝑦))
31	25, 30	impbid2 229	. . . . 5 ⊢ ((𝑥 ∈ Q ∧ 𝑦 ∈ Q) → (𝑥 = 𝑦 ↔ ((1st ‘𝑥) ·N (2nd ‘𝑦)) = ((1st ‘𝑦) ·N (2nd ‘𝑥))))
32		ordpinq 11055	. . . . . 6 ⊢ ((𝑦 ∈ Q ∧ 𝑥 ∈ Q) → (𝑦 <Q 𝑥 ↔ ((1st ‘𝑦) ·N (2nd ‘𝑥)) <N ((1st ‘𝑥) ·N (2nd ‘𝑦))))
33	32	ancoms 464	. . . . 5 ⊢ ((𝑥 ∈ Q ∧ 𝑦 ∈ Q) → (𝑦 <Q 𝑥 ↔ ((1st ‘𝑦) ·N (2nd ‘𝑥)) <N ((1st ‘𝑥) ·N (2nd ‘𝑦))))
34	31, 33	orbi12d 932	. . . 4 ⊢ ((𝑥 ∈ Q ∧ 𝑦 ∈ Q) → ((𝑥 = 𝑦 ∨ 𝑦 <Q 𝑥) ↔ (((1st ‘𝑥) ·N (2nd ‘𝑦)) = ((1st ‘𝑦) ·N (2nd ‘𝑥)) ∨ ((1st ‘𝑦) ·N (2nd ‘𝑥)) <N ((1st ‘𝑥) ·N (2nd ‘𝑦)))))
35	34	notbid 321	. . 3 ⊢ ((𝑥 ∈ Q ∧ 𝑦 ∈ Q) → (¬ (𝑥 = 𝑦 ∨ 𝑦 <Q 𝑥) ↔ ¬ (((1st ‘𝑥) ·N (2nd ‘𝑦)) = ((1st ‘𝑦) ·N (2nd ‘𝑥)) ∨ ((1st ‘𝑦) ·N (2nd ‘𝑥)) <N ((1st ‘𝑥) ·N (2nd ‘𝑦)))))
36	20, 21, 35	3bitr4d 314	. 2 ⊢ ((𝑥 ∈ Q ∧ 𝑦 ∈ Q) → (𝑥 <Q 𝑦 ↔ ¬ (𝑥 = 𝑦 ∨ 𝑦 <Q 𝑥)))
37	21	3adant3 1150	. . . . . 6 ⊢ ((𝑥 ∈ Q ∧ 𝑦 ∈ Q ∧ 𝑧 ∈ Q) → (𝑥 <Q 𝑦 ↔ ((1st ‘𝑥) ·N (2nd ‘𝑦)) <N ((1st ‘𝑦) ·N (2nd ‘𝑥))))
38		elpqn 11037	. . . . . . . 8 ⊢ (𝑧 ∈ Q → 𝑧 ∈ (N × N))
39	38	3ad2ant3 1153	. . . . . . 7 ⊢ ((𝑥 ∈ Q ∧ 𝑦 ∈ Q ∧ 𝑧 ∈ Q) → 𝑧 ∈ (N × N))
40		xp2nd 8058	. . . . . . 7 ⊢ (𝑧 ∈ (N × N) → (2nd ‘𝑧) ∈ N)
41		ltmpi 11016	. . . . . . 7 ⊢ ((2nd ‘𝑧) ∈ N → (((1st ‘𝑥) ·N (2nd ‘𝑦)) <N ((1st ‘𝑦) ·N (2nd ‘𝑥)) ↔ ((2nd ‘𝑧) ·N ((1st ‘𝑥) ·N (2nd ‘𝑦))) <N ((2nd ‘𝑧) ·N ((1st ‘𝑦) ·N (2nd ‘𝑥)))))
42	39, 40, 41	3syl 19	. . . . . 6 ⊢ ((𝑥 ∈ Q ∧ 𝑦 ∈ Q ∧ 𝑧 ∈ Q) → (((1st ‘𝑥) ·N (2nd ‘𝑦)) <N ((1st ‘𝑦) ·N (2nd ‘𝑥)) ↔ ((2nd ‘𝑧) ·N ((1st ‘𝑥) ·N (2nd ‘𝑦))) <N ((2nd ‘𝑧) ·N ((1st ‘𝑦) ·N (2nd ‘𝑥)))))
43	37, 42	bitrd 282	. . . . 5 ⊢ ((𝑥 ∈ Q ∧ 𝑦 ∈ Q ∧ 𝑧 ∈ Q) → (𝑥 <Q 𝑦 ↔ ((2nd ‘𝑧) ·N ((1st ‘𝑥) ·N (2nd ‘𝑦))) <N ((2nd ‘𝑧) ·N ((1st ‘𝑦) ·N (2nd ‘𝑥)))))
44		ordpinq 11055	. . . . . . 7 ⊢ ((𝑦 ∈ Q ∧ 𝑧 ∈ Q) → (𝑦 <Q 𝑧 ↔ ((1st ‘𝑦) ·N (2nd ‘𝑧)) <N ((1st ‘𝑧) ·N (2nd ‘𝑦))))
45	44	3adant1 1148	. . . . . 6 ⊢ ((𝑥 ∈ Q ∧ 𝑦 ∈ Q ∧ 𝑧 ∈ Q) → (𝑦 <Q 𝑧 ↔ ((1st ‘𝑦) ·N (2nd ‘𝑧)) <N ((1st ‘𝑧) ·N (2nd ‘𝑦))))
46	1	3ad2ant1 1151	. . . . . . 7 ⊢ ((𝑥 ∈ Q ∧ 𝑦 ∈ Q ∧ 𝑧 ∈ Q) → 𝑥 ∈ (N × N))
47		ltmpi 11016	. . . . . . 7 ⊢ ((2nd ‘𝑥) ∈ N → (((1st ‘𝑦) ·N (2nd ‘𝑧)) <N ((1st ‘𝑧) ·N (2nd ‘𝑦)) ↔ ((2nd ‘𝑥) ·N ((1st ‘𝑦) ·N (2nd ‘𝑧))) <N ((2nd ‘𝑥) ·N ((1st ‘𝑧) ·N (2nd ‘𝑦)))))
48	46, 13, 47	3syl 19	. . . . . 6 ⊢ ((𝑥 ∈ Q ∧ 𝑦 ∈ Q ∧ 𝑧 ∈ Q) → (((1st ‘𝑦) ·N (2nd ‘𝑧)) <N ((1st ‘𝑧) ·N (2nd ‘𝑦)) ↔ ((2nd ‘𝑥) ·N ((1st ‘𝑦) ·N (2nd ‘𝑧))) <N ((2nd ‘𝑥) ·N ((1st ‘𝑧) ·N (2nd ‘𝑦)))))
49	45, 48	bitrd 282	. . . . 5 ⊢ ((𝑥 ∈ Q ∧ 𝑦 ∈ Q ∧ 𝑧 ∈ Q) → (𝑦 <Q 𝑧 ↔ ((2nd ‘𝑥) ·N ((1st ‘𝑦) ·N (2nd ‘𝑧))) <N ((2nd ‘𝑥) ·N ((1st ‘𝑧) ·N (2nd ‘𝑦)))))
50	43, 49	anbi12d 644	. . . 4 ⊢ ((𝑥 ∈ Q ∧ 𝑦 ∈ Q ∧ 𝑧 ∈ Q) → ((𝑥 <Q 𝑦 ∧ 𝑦 <Q 𝑧) ↔ (((2nd ‘𝑧) ·N ((1st ‘𝑥) ·N (2nd ‘𝑦))) <N ((2nd ‘𝑧) ·N ((1st ‘𝑦) ·N (2nd ‘𝑥))) ∧ ((2nd ‘𝑥) ·N ((1st ‘𝑦) ·N (2nd ‘𝑧))) <N ((2nd ‘𝑥) ·N ((1st ‘𝑧) ·N (2nd ‘𝑦))))))
51		fvex 6921	. . . . . . 7 ⊢ (2nd ‘𝑥) ∈ V
52		fvex 6921	. . . . . . 7 ⊢ (1st ‘𝑦) ∈ V
53		fvex 6921	. . . . . . 7 ⊢ (2nd ‘𝑧) ∈ V
54		mulcompi 11008	. . . . . . 7 ⊢ (𝑟 ·N 𝑠) = (𝑠 ·N 𝑟)
55		mulasspi 11009	. . . . . . 7 ⊢ ((𝑟 ·N 𝑠) ·N 𝑡) = (𝑟 ·N (𝑠 ·N 𝑡))
56	51, 52, 53, 54, 55	caov13 7675	. . . . . 6 ⊢ ((2nd ‘𝑥) ·N ((1st ‘𝑦) ·N (2nd ‘𝑧))) = ((2nd ‘𝑧) ·N ((1st ‘𝑦) ·N (2nd ‘𝑥)))
57		fvex 6921	. . . . . . 7 ⊢ (1st ‘𝑧) ∈ V
58		fvex 6921	. . . . . . 7 ⊢ (2nd ‘𝑦) ∈ V
59	51, 57, 58, 54, 55	caov13 7675	. . . . . 6 ⊢ ((2nd ‘𝑥) ·N ((1st ‘𝑧) ·N (2nd ‘𝑦))) = ((2nd ‘𝑦) ·N ((1st ‘𝑧) ·N (2nd ‘𝑥)))
60	56, 59	breq12i 5115	. . . . 5 ⊢ (((2nd ‘𝑥) ·N ((1st ‘𝑦) ·N (2nd ‘𝑧))) <N ((2nd ‘𝑥) ·N ((1st ‘𝑧) ·N (2nd ‘𝑦))) ↔ ((2nd ‘𝑧) ·N ((1st ‘𝑦) ·N (2nd ‘𝑥))) <N ((2nd ‘𝑦) ·N ((1st ‘𝑧) ·N (2nd ‘𝑥))))
61		fvex 6921	. . . . . . 7 ⊢ (1st ‘𝑥) ∈ V
62	53, 61, 58, 54, 55	caov13 7675	. . . . . 6 ⊢ ((2nd ‘𝑧) ·N ((1st ‘𝑥) ·N (2nd ‘𝑦))) = ((2nd ‘𝑦) ·N ((1st ‘𝑥) ·N (2nd ‘𝑧)))
63		ltrelpi 11001	. . . . . . 7 ⊢ <N ⊆ (N × N)
64	17, 63	sotri 6133	. . . . . 6 ⊢ ((((2nd ‘𝑧) ·N ((1st ‘𝑥) ·N (2nd ‘𝑦))) <N ((2nd ‘𝑧) ·N ((1st ‘𝑦) ·N (2nd ‘𝑥))) ∧ ((2nd ‘𝑧) ·N ((1st ‘𝑦) ·N (2nd ‘𝑥))) <N ((2nd ‘𝑦) ·N ((1st ‘𝑧) ·N (2nd ‘𝑥)))) → ((2nd ‘𝑧) ·N ((1st ‘𝑥) ·N (2nd ‘𝑦))) <N ((2nd ‘𝑦) ·N ((1st ‘𝑧) ·N (2nd ‘𝑥))))
65	62, 64	eqbrtrrid 5144	. . . . 5 ⊢ ((((2nd ‘𝑧) ·N ((1st ‘𝑥) ·N (2nd ‘𝑦))) <N ((2nd ‘𝑧) ·N ((1st ‘𝑦) ·N (2nd ‘𝑥))) ∧ ((2nd ‘𝑧) ·N ((1st ‘𝑦) ·N (2nd ‘𝑥))) <N ((2nd ‘𝑦) ·N ((1st ‘𝑧) ·N (2nd ‘𝑥)))) → ((2nd ‘𝑦) ·N ((1st ‘𝑥) ·N (2nd ‘𝑧))) <N ((2nd ‘𝑦) ·N ((1st ‘𝑧) ·N (2nd ‘𝑥))))
66	60, 65	sylan2b 606	. . . 4 ⊢ ((((2nd ‘𝑧) ·N ((1st ‘𝑥) ·N (2nd ‘𝑦))) <N ((2nd ‘𝑧) ·N ((1st ‘𝑦) ·N (2nd ‘𝑥))) ∧ ((2nd ‘𝑥) ·N ((1st ‘𝑦) ·N (2nd ‘𝑧))) <N ((2nd ‘𝑥) ·N ((1st ‘𝑧) ·N (2nd ‘𝑦)))) → ((2nd ‘𝑦) ·N ((1st ‘𝑥) ·N (2nd ‘𝑧))) <N ((2nd ‘𝑦) ·N ((1st ‘𝑧) ·N (2nd ‘𝑥))))
67	50, 66	biimtrdi 256	. . 3 ⊢ ((𝑥 ∈ Q ∧ 𝑦 ∈ Q ∧ 𝑧 ∈ Q) → ((𝑥 <Q 𝑦 ∧ 𝑦 <Q 𝑧) → ((2nd ‘𝑦) ·N ((1st ‘𝑥) ·N (2nd ‘𝑧))) <N ((2nd ‘𝑦) ·N ((1st ‘𝑧) ·N (2nd ‘𝑥)))))
68		ordpinq 11055	. . . . 5 ⊢ ((𝑥 ∈ Q ∧ 𝑧 ∈ Q) → (𝑥 <Q 𝑧 ↔ ((1st ‘𝑥) ·N (2nd ‘𝑧)) <N ((1st ‘𝑧) ·N (2nd ‘𝑥))))
69	68	3adant2 1149	. . . 4 ⊢ ((𝑥 ∈ Q ∧ 𝑦 ∈ Q ∧ 𝑧 ∈ Q) → (𝑥 <Q 𝑧 ↔ ((1st ‘𝑥) ·N (2nd ‘𝑧)) <N ((1st ‘𝑧) ·N (2nd ‘𝑥))))
70	5	3ad2ant2 1152	. . . . 5 ⊢ ((𝑥 ∈ Q ∧ 𝑦 ∈ Q ∧ 𝑧 ∈ Q) → 𝑦 ∈ (N × N))
71		ltmpi 11016	. . . . 5 ⊢ ((2nd ‘𝑦) ∈ N → (((1st ‘𝑥) ·N (2nd ‘𝑧)) <N ((1st ‘𝑧) ·N (2nd ‘𝑥)) ↔ ((2nd ‘𝑦) ·N ((1st ‘𝑥) ·N (2nd ‘𝑧))) <N ((2nd ‘𝑦) ·N ((1st ‘𝑧) ·N (2nd ‘𝑥)))))
72	70, 7, 71	3syl 19	. . . 4 ⊢ ((𝑥 ∈ Q ∧ 𝑦 ∈ Q ∧ 𝑧 ∈ Q) → (((1st ‘𝑥) ·N (2nd ‘𝑧)) <N ((1st ‘𝑧) ·N (2nd ‘𝑥)) ↔ ((2nd ‘𝑦) ·N ((1st ‘𝑥) ·N (2nd ‘𝑧))) <N ((2nd ‘𝑦) ·N ((1st ‘𝑧) ·N (2nd ‘𝑥)))))
73	69, 72	bitrd 282	. . 3 ⊢ ((𝑥 ∈ Q ∧ 𝑦 ∈ Q ∧ 𝑧 ∈ Q) → (𝑥 <Q 𝑧 ↔ ((2nd ‘𝑦) ·N ((1st ‘𝑥) ·N (2nd ‘𝑧))) <N ((2nd ‘𝑦) ·N ((1st ‘𝑧) ·N (2nd ‘𝑥)))))
74	67, 73	sylibrd 262	. 2 ⊢ ((𝑥 ∈ Q ∧ 𝑦 ∈ Q ∧ 𝑧 ∈ Q) → ((𝑥 <Q 𝑦 ∧ 𝑦 <Q 𝑧) → 𝑥 <Q 𝑧))
75	36, 74	isso2i 5600	1 ⊢ <Q Or Q

This proof depends on syntax axioms:  ¬ wn 3   ↔ wb 209   ∧ wa 401   ∨ wo 861   ∧ w3a 1103   = wceq 1570   ∈ wcel 2147   class class class wbr 5106   Or wor 5562   × cxp 5653  ‘cfv 6558  (class class class)co 7443  1^st c1st 8023  2^nd c2nd 8024  Ncnpi 10956   ·_N cmi 10958   <_N clti 10959   ~_Q ceq 10963  Qcnq 10964   <_Q cltq 10970

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5252  ax-nul 5263  ax-pr 5395  ax-un 7775

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-iun 4956  df-br 5107  df-opab 5171  df-mpt 5190  df-tr 5216  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6322  df-ord 6383  df-on 6384  df-lim 6385  df-suc 6386  df-iota 6512  df-fun 6560  df-fn 6561  df-f 6562  df-f1 6563  df-fo 6564  df-f1o 6565  df-fv 6566  df-ov 7446  df-oprab 7447  df-mpo 7448  df-om 7902  df-1st 8025  df-2nd 8026  df-frecs 8327  df-wrecs 8358  df-recs 8407  df-rdg 8446  df-oadd 8508  df-omul 8509  df-er 8745  df-ni 10984  df-mi 10986  df-lti 10987  df-ltpq 11022  df-enq 11023  df-nq 11024  df-ltnq 11030

This theorem is used by:  ltbtwnnq  11090  prub  11106  npomex  11108  genpnnp  11117  nqpr  11126  distrlem4pr  11138  prlem934  11145  ltexprlem4  11151  reclem2pr  11160  reclem4pr  11162