Theorem ltsonq 10380

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 10336	. . . . . . 7 ⊢ (𝑥 ∈ Q → 𝑥 ∈ (N × N))
2	1	adantr 484	. . . . . 6 ⊢ ((𝑥 ∈ Q ∧ 𝑦 ∈ Q) → 𝑥 ∈ (N × N))
3		xp1st 7703	. . . . . 6 ⊢ (𝑥 ∈ (N × N) → (1st ‘𝑥) ∈ N)
4	2, 3	syl 17	. . . . 5 ⊢ ((𝑥 ∈ Q ∧ 𝑦 ∈ Q) → (1st ‘𝑥) ∈ N)
5		elpqn 10336	. . . . . . 7 ⊢ (𝑦 ∈ Q → 𝑦 ∈ (N × N))
6	5	adantl 485	. . . . . 6 ⊢ ((𝑥 ∈ Q ∧ 𝑦 ∈ Q) → 𝑦 ∈ (N × N))
7		xp2nd 7704	. . . . . 6 ⊢ (𝑦 ∈ (N × N) → (2nd ‘𝑦) ∈ N)
8	6, 7	syl 17	. . . . 5 ⊢ ((𝑥 ∈ Q ∧ 𝑦 ∈ Q) → (2nd ‘𝑦) ∈ N)
9		mulclpi 10304	. . . . 5 ⊢ (((1st ‘𝑥) ∈ N ∧ (2nd ‘𝑦) ∈ N) → ((1st ‘𝑥) ·N (2nd ‘𝑦)) ∈ N)
10	4, 8, 9	syl2anc 587	. . . 4 ⊢ ((𝑥 ∈ Q ∧ 𝑦 ∈ Q) → ((1st ‘𝑥) ·N (2nd ‘𝑦)) ∈ N)
11		xp1st 7703	. . . . . 6 ⊢ (𝑦 ∈ (N × N) → (1st ‘𝑦) ∈ N)
12	6, 11	syl 17	. . . . 5 ⊢ ((𝑥 ∈ Q ∧ 𝑦 ∈ Q) → (1st ‘𝑦) ∈ N)
13		xp2nd 7704	. . . . . 6 ⊢ (𝑥 ∈ (N × N) → (2nd ‘𝑥) ∈ N)
14	2, 13	syl 17	. . . . 5 ⊢ ((𝑥 ∈ Q ∧ 𝑦 ∈ Q) → (2nd ‘𝑥) ∈ N)
15		mulclpi 10304	. . . . 5 ⊢ (((1st ‘𝑦) ∈ N ∧ (2nd ‘𝑥) ∈ N) → ((1st ‘𝑦) ·N (2nd ‘𝑥)) ∈ N)
16	12, 14, 15	syl2anc 587	. . . 4 ⊢ ((𝑥 ∈ Q ∧ 𝑦 ∈ Q) → ((1st ‘𝑦) ·N (2nd ‘𝑥)) ∈ N)
17		ltsopi 10299	. . . . 5 ⊢ <N Or N
18		sotric 5465	. . . . 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 689	. . . 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 587	. . 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 10354	. . 3 ⊢ ((𝑥 ∈ Q ∧ 𝑦 ∈ Q) → (𝑥 <Q 𝑦 ↔ ((1st ‘𝑥) ·N (2nd ‘𝑦)) <N ((1st ‘𝑦) ·N (2nd ‘𝑥))))
22		fveq2 6645	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (1st ‘𝑥) = (1st ‘𝑦))
23		fveq2 6645	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (2nd ‘𝑥) = (2nd ‘𝑦))
24	23	eqcomd 2804	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (2nd ‘𝑦) = (2nd ‘𝑥))
25	22, 24	oveq12d 7153	. . . . . 6 ⊢ (𝑥 = 𝑦 → ((1st ‘𝑥) ·N (2nd ‘𝑦)) = ((1st ‘𝑦) ·N (2nd ‘𝑥)))
26		enqbreq2 10331	. . . . . . . 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 598	. . . . . . 7 ⊢ ((𝑥 ∈ Q ∧ 𝑦 ∈ Q) → (𝑥 ~Q 𝑦 ↔ ((1st ‘𝑥) ·N (2nd ‘𝑦)) = ((1st ‘𝑦) ·N (2nd ‘𝑥))))
28		enqeq 10345	. . . . . . . 8 ⊢ ((𝑥 ∈ Q ∧ 𝑦 ∈ Q ∧ 𝑥 ~Q 𝑦) → 𝑥 = 𝑦)
29	28	3expia 1118	. . . . . . 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 10354	. . . . . 6 ⊢ ((𝑦 ∈ Q ∧ 𝑥 ∈ Q) → (𝑦 <Q 𝑥 ↔ ((1st ‘𝑦) ·N (2nd ‘𝑥)) <N ((1st ‘𝑥) ·N (2nd ‘𝑦))))
33	32	ancoms 462	. . . . 5 ⊢ ((𝑥 ∈ Q ∧ 𝑦 ∈ Q) → (𝑦 <Q 𝑥 ↔ ((1st ‘𝑦) ·N (2nd ‘𝑥)) <N ((1st ‘𝑥) ·N (2nd ‘𝑦))))
34	31, 33	orbi12d 916	. . . 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 1129	. . . . . 6 ⊢ ((𝑥 ∈ Q ∧ 𝑦 ∈ Q ∧ 𝑧 ∈ Q) → (𝑥 <Q 𝑦 ↔ ((1st ‘𝑥) ·N (2nd ‘𝑦)) <N ((1st ‘𝑦) ·N (2nd ‘𝑥))))
38		elpqn 10336	. . . . . . . 8 ⊢ (𝑧 ∈ Q → 𝑧 ∈ (N × N))
39	38	3ad2ant3 1132	. . . . . . 7 ⊢ ((𝑥 ∈ Q ∧ 𝑦 ∈ Q ∧ 𝑧 ∈ Q) → 𝑧 ∈ (N × N))
40		xp2nd 7704	. . . . . . 7 ⊢ (𝑧 ∈ (N × N) → (2nd ‘𝑧) ∈ N)
41		ltmpi 10315	. . . . . . 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 18	. . . . . 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 10354	. . . . . . 7 ⊢ ((𝑦 ∈ Q ∧ 𝑧 ∈ Q) → (𝑦 <Q 𝑧 ↔ ((1st ‘𝑦) ·N (2nd ‘𝑧)) <N ((1st ‘𝑧) ·N (2nd ‘𝑦))))
45	44	3adant1 1127	. . . . . 6 ⊢ ((𝑥 ∈ Q ∧ 𝑦 ∈ Q ∧ 𝑧 ∈ Q) → (𝑦 <Q 𝑧 ↔ ((1st ‘𝑦) ·N (2nd ‘𝑧)) <N ((1st ‘𝑧) ·N (2nd ‘𝑦))))
46	1	3ad2ant1 1130	. . . . . . 7 ⊢ ((𝑥 ∈ Q ∧ 𝑦 ∈ Q ∧ 𝑧 ∈ Q) → 𝑥 ∈ (N × N))
47		ltmpi 10315	. . . . . . 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 18	. . . . . 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 633	. . . 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 6658	. . . . . . 7 ⊢ (2nd ‘𝑥) ∈ V
52		fvex 6658	. . . . . . 7 ⊢ (1st ‘𝑦) ∈ V
53		fvex 6658	. . . . . . 7 ⊢ (2nd ‘𝑧) ∈ V
54		mulcompi 10307	. . . . . . 7 ⊢ (𝑟 ·N 𝑠) = (𝑠 ·N 𝑟)
55		mulasspi 10308	. . . . . . 7 ⊢ ((𝑟 ·N 𝑠) ·N 𝑡) = (𝑟 ·N (𝑠 ·N 𝑡))
56	51, 52, 53, 54, 55	caov13 7358	. . . . . 6 ⊢ ((2nd ‘𝑥) ·N ((1st ‘𝑦) ·N (2nd ‘𝑧))) = ((2nd ‘𝑧) ·N ((1st ‘𝑦) ·N (2nd ‘𝑥)))
57		fvex 6658	. . . . . . 7 ⊢ (1st ‘𝑧) ∈ V
58		fvex 6658	. . . . . . 7 ⊢ (2nd ‘𝑦) ∈ V
59	51, 57, 58, 54, 55	caov13 7358	. . . . . 6 ⊢ ((2nd ‘𝑥) ·N ((1st ‘𝑧) ·N (2nd ‘𝑦))) = ((2nd ‘𝑦) ·N ((1st ‘𝑧) ·N (2nd ‘𝑥)))
60	56, 59	breq12i 5039	. . . . 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 6658	. . . . . . 7 ⊢ (1st ‘𝑥) ∈ V
62	53, 61, 58, 54, 55	caov13 7358	. . . . . 6 ⊢ ((2nd ‘𝑧) ·N ((1st ‘𝑥) ·N (2nd ‘𝑦))) = ((2nd ‘𝑦) ·N ((1st ‘𝑥) ·N (2nd ‘𝑧)))
63		ltrelpi 10300	. . . . . . 7 ⊢ <N ⊆ (N × N)
64	17, 63	sotri 5954	. . . . . 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 5066	. . . . 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 596	. . . 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	syl6bi 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 10354	. . . . 5 ⊢ ((𝑥 ∈ Q ∧ 𝑧 ∈ Q) → (𝑥 <Q 𝑧 ↔ ((1st ‘𝑥) ·N (2nd ‘𝑧)) <N ((1st ‘𝑧) ·N (2nd ‘𝑥))))
69	68	3adant2 1128	. . . 4 ⊢ ((𝑥 ∈ Q ∧ 𝑦 ∈ Q ∧ 𝑧 ∈ Q) → (𝑥 <Q 𝑧 ↔ ((1st ‘𝑥) ·N (2nd ‘𝑧)) <N ((1st ‘𝑧) ·N (2nd ‘𝑥))))
70	5	3ad2ant2 1131	. . . . 5 ⊢ ((𝑥 ∈ Q ∧ 𝑦 ∈ Q ∧ 𝑧 ∈ Q) → 𝑦 ∈ (N × N))
71		ltmpi 10315	. . . . 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 18	. . . 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 5472	1 ⊢ <Q Or Q

Syntax hints:  ¬ wn 3   ↔ wb 209   ∧ wa 399   ∨ wo 844   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111   class class class wbr 5030   Or wor 5437   × cxp 5517  ‘cfv 6324  (class class class)co 7135  1^st c1st 7669  2^nd c2nd 7670  Ncnpi 10255   ·_N cmi 10257   <_N clti 10258   ~_Q ceq 10262  Qcnq 10263   <_Q cltq 10269

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rmo 3114  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4801  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-pred 6116  df-ord 6162  df-on 6163  df-lim 6164  df-suc 6165  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-ov 7138  df-oprab 7139  df-mpo 7140  df-om 7561  df-1st 7671  df-2nd 7672  df-wrecs 7930  df-recs 7991  df-rdg 8029  df-oadd 8089  df-omul 8090  df-er 8272  df-ni 10283  df-mi 10285  df-lti 10286  df-ltpq 10321  df-enq 10322  df-nq 10323  df-ltnq 10329

This theorem is referenced by:  ltbtwnnq  10389  prub  10405  npomex  10407  genpnnp  10416  nqpr  10425  distrlem4pr  10437  prlem934  10444  ltexprlem4  10450  reclem2pr  10459  reclem4pr  10461