Theorem ltsonq 10393

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 10349	. . . . . . 7 ⊢ (𝑥 ∈ Q → 𝑥 ∈ (N × N))
2	1	adantr 483	. . . . . 6 ⊢ ((𝑥 ∈ Q ∧ 𝑦 ∈ Q) → 𝑥 ∈ (N × N))
3		xp1st 7723	. . . . . 6 ⊢ (𝑥 ∈ (N × N) → (1st ‘𝑥) ∈ N)
4	2, 3	syl 17	. . . . 5 ⊢ ((𝑥 ∈ Q ∧ 𝑦 ∈ Q) → (1st ‘𝑥) ∈ N)
5		elpqn 10349	. . . . . . 7 ⊢ (𝑦 ∈ Q → 𝑦 ∈ (N × N))
6	5	adantl 484	. . . . . 6 ⊢ ((𝑥 ∈ Q ∧ 𝑦 ∈ Q) → 𝑦 ∈ (N × N))
7		xp2nd 7724	. . . . . 6 ⊢ (𝑦 ∈ (N × N) → (2nd ‘𝑦) ∈ N)
8	6, 7	syl 17	. . . . 5 ⊢ ((𝑥 ∈ Q ∧ 𝑦 ∈ Q) → (2nd ‘𝑦) ∈ N)
9		mulclpi 10317	. . . . 5 ⊢ (((1st ‘𝑥) ∈ N ∧ (2nd ‘𝑦) ∈ N) → ((1st ‘𝑥) ·N (2nd ‘𝑦)) ∈ N)
10	4, 8, 9	syl2anc 586	. . . 4 ⊢ ((𝑥 ∈ Q ∧ 𝑦 ∈ Q) → ((1st ‘𝑥) ·N (2nd ‘𝑦)) ∈ N)
11		xp1st 7723	. . . . . 6 ⊢ (𝑦 ∈ (N × N) → (1st ‘𝑦) ∈ N)
12	6, 11	syl 17	. . . . 5 ⊢ ((𝑥 ∈ Q ∧ 𝑦 ∈ Q) → (1st ‘𝑦) ∈ N)
13		xp2nd 7724	. . . . . 6 ⊢ (𝑥 ∈ (N × N) → (2nd ‘𝑥) ∈ N)
14	2, 13	syl 17	. . . . 5 ⊢ ((𝑥 ∈ Q ∧ 𝑦 ∈ Q) → (2nd ‘𝑥) ∈ N)
15		mulclpi 10317	. . . . 5 ⊢ (((1st ‘𝑦) ∈ N ∧ (2nd ‘𝑥) ∈ N) → ((1st ‘𝑦) ·N (2nd ‘𝑥)) ∈ N)
16	12, 14, 15	syl2anc 586	. . . 4 ⊢ ((𝑥 ∈ Q ∧ 𝑦 ∈ Q) → ((1st ‘𝑦) ·N (2nd ‘𝑥)) ∈ N)
17		ltsopi 10312	. . . . 5 ⊢ <N Or N
18		sotric 5503	. . . . 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 688	. . . 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 586	. . 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 10367	. . 3 ⊢ ((𝑥 ∈ Q ∧ 𝑦 ∈ Q) → (𝑥 <Q 𝑦 ↔ ((1st ‘𝑥) ·N (2nd ‘𝑦)) <N ((1st ‘𝑦) ·N (2nd ‘𝑥))))
22		fveq2 6672	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (1st ‘𝑥) = (1st ‘𝑦))
23		fveq2 6672	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (2nd ‘𝑥) = (2nd ‘𝑦))
24	23	eqcomd 2829	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (2nd ‘𝑦) = (2nd ‘𝑥))
25	22, 24	oveq12d 7176	. . . . . 6 ⊢ (𝑥 = 𝑦 → ((1st ‘𝑥) ·N (2nd ‘𝑦)) = ((1st ‘𝑦) ·N (2nd ‘𝑥)))
26		enqbreq2 10344	. . . . . . . 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 597	. . . . . . 7 ⊢ ((𝑥 ∈ Q ∧ 𝑦 ∈ Q) → (𝑥 ~Q 𝑦 ↔ ((1st ‘𝑥) ·N (2nd ‘𝑦)) = ((1st ‘𝑦) ·N (2nd ‘𝑥))))
28		enqeq 10358	. . . . . . . 8 ⊢ ((𝑥 ∈ Q ∧ 𝑦 ∈ Q ∧ 𝑥 ~Q 𝑦) → 𝑥 = 𝑦)
29	28	3expia 1117	. . . . . . 7 ⊢ ((𝑥 ∈ Q ∧ 𝑦 ∈ Q) → (𝑥 ~Q 𝑦 → 𝑥 = 𝑦))
30	27, 29	sylbird 262	. . . . . 6 ⊢ ((𝑥 ∈ Q ∧ 𝑦 ∈ Q) → (((1st ‘𝑥) ·N (2nd ‘𝑦)) = ((1st ‘𝑦) ·N (2nd ‘𝑥)) → 𝑥 = 𝑦))
31	25, 30	impbid2 228	. . . . 5 ⊢ ((𝑥 ∈ Q ∧ 𝑦 ∈ Q) → (𝑥 = 𝑦 ↔ ((1st ‘𝑥) ·N (2nd ‘𝑦)) = ((1st ‘𝑦) ·N (2nd ‘𝑥))))
32		ordpinq 10367	. . . . . 6 ⊢ ((𝑦 ∈ Q ∧ 𝑥 ∈ Q) → (𝑦 <Q 𝑥 ↔ ((1st ‘𝑦) ·N (2nd ‘𝑥)) <N ((1st ‘𝑥) ·N (2nd ‘𝑦))))
33	32	ancoms 461	. . . . 5 ⊢ ((𝑥 ∈ Q ∧ 𝑦 ∈ Q) → (𝑦 <Q 𝑥 ↔ ((1st ‘𝑦) ·N (2nd ‘𝑥)) <N ((1st ‘𝑥) ·N (2nd ‘𝑦))))
34	31, 33	orbi12d 915	. . . 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 320	. . 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 313	. 2 ⊢ ((𝑥 ∈ Q ∧ 𝑦 ∈ Q) → (𝑥 <Q 𝑦 ↔ ¬ (𝑥 = 𝑦 ∨ 𝑦 <Q 𝑥)))
37	21	3adant3 1128	. . . . . 6 ⊢ ((𝑥 ∈ Q ∧ 𝑦 ∈ Q ∧ 𝑧 ∈ Q) → (𝑥 <Q 𝑦 ↔ ((1st ‘𝑥) ·N (2nd ‘𝑦)) <N ((1st ‘𝑦) ·N (2nd ‘𝑥))))
38		elpqn 10349	. . . . . . . 8 ⊢ (𝑧 ∈ Q → 𝑧 ∈ (N × N))
39	38	3ad2ant3 1131	. . . . . . 7 ⊢ ((𝑥 ∈ Q ∧ 𝑦 ∈ Q ∧ 𝑧 ∈ Q) → 𝑧 ∈ (N × N))
40		xp2nd 7724	. . . . . . 7 ⊢ (𝑧 ∈ (N × N) → (2nd ‘𝑧) ∈ N)
41		ltmpi 10328	. . . . . . 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 281	. . . . 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 10367	. . . . . . 7 ⊢ ((𝑦 ∈ Q ∧ 𝑧 ∈ Q) → (𝑦 <Q 𝑧 ↔ ((1st ‘𝑦) ·N (2nd ‘𝑧)) <N ((1st ‘𝑧) ·N (2nd ‘𝑦))))
45	44	3adant1 1126	. . . . . 6 ⊢ ((𝑥 ∈ Q ∧ 𝑦 ∈ Q ∧ 𝑧 ∈ Q) → (𝑦 <Q 𝑧 ↔ ((1st ‘𝑦) ·N (2nd ‘𝑧)) <N ((1st ‘𝑧) ·N (2nd ‘𝑦))))
46	1	3ad2ant1 1129	. . . . . . 7 ⊢ ((𝑥 ∈ Q ∧ 𝑦 ∈ Q ∧ 𝑧 ∈ Q) → 𝑥 ∈ (N × N))
47		ltmpi 10328	. . . . . . 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 281	. . . . 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 632	. . . 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 6685	. . . . . . 7 ⊢ (2nd ‘𝑥) ∈ V
52		fvex 6685	. . . . . . 7 ⊢ (1st ‘𝑦) ∈ V
53		fvex 6685	. . . . . . 7 ⊢ (2nd ‘𝑧) ∈ V
54		mulcompi 10320	. . . . . . 7 ⊢ (𝑟 ·N 𝑠) = (𝑠 ·N 𝑟)
55		mulasspi 10321	. . . . . . 7 ⊢ ((𝑟 ·N 𝑠) ·N 𝑡) = (𝑟 ·N (𝑠 ·N 𝑡))
56	51, 52, 53, 54, 55	caov13 7380	. . . . . 6 ⊢ ((2nd ‘𝑥) ·N ((1st ‘𝑦) ·N (2nd ‘𝑧))) = ((2nd ‘𝑧) ·N ((1st ‘𝑦) ·N (2nd ‘𝑥)))
57		fvex 6685	. . . . . . 7 ⊢ (1st ‘𝑧) ∈ V
58		fvex 6685	. . . . . . 7 ⊢ (2nd ‘𝑦) ∈ V
59	51, 57, 58, 54, 55	caov13 7380	. . . . . 6 ⊢ ((2nd ‘𝑥) ·N ((1st ‘𝑧) ·N (2nd ‘𝑦))) = ((2nd ‘𝑦) ·N ((1st ‘𝑧) ·N (2nd ‘𝑥)))
60	56, 59	breq12i 5077	. . . . 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 6685	. . . . . . 7 ⊢ (1st ‘𝑥) ∈ V
62	53, 61, 58, 54, 55	caov13 7380	. . . . . 6 ⊢ ((2nd ‘𝑧) ·N ((1st ‘𝑥) ·N (2nd ‘𝑦))) = ((2nd ‘𝑦) ·N ((1st ‘𝑥) ·N (2nd ‘𝑧)))
63		ltrelpi 10313	. . . . . . 7 ⊢ <N ⊆ (N × N)
64	17, 63	sotri 5989	. . . . . 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 5104	. . . . 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 595	. . . 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 255	. . 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 10367	. . . . 5 ⊢ ((𝑥 ∈ Q ∧ 𝑧 ∈ Q) → (𝑥 <Q 𝑧 ↔ ((1st ‘𝑥) ·N (2nd ‘𝑧)) <N ((1st ‘𝑧) ·N (2nd ‘𝑥))))
69	68	3adant2 1127	. . . 4 ⊢ ((𝑥 ∈ Q ∧ 𝑦 ∈ Q ∧ 𝑧 ∈ Q) → (𝑥 <Q 𝑧 ↔ ((1st ‘𝑥) ·N (2nd ‘𝑧)) <N ((1st ‘𝑧) ·N (2nd ‘𝑥))))
70	5	3ad2ant2 1130	. . . . 5 ⊢ ((𝑥 ∈ Q ∧ 𝑦 ∈ Q ∧ 𝑧 ∈ Q) → 𝑦 ∈ (N × N))
71		ltmpi 10328	. . . . 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 281	. . 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 261	. 2 ⊢ ((𝑥 ∈ Q ∧ 𝑦 ∈ Q ∧ 𝑧 ∈ Q) → ((𝑥 <Q 𝑦 ∧ 𝑦 <Q 𝑧) → 𝑥 <Q 𝑧))
75	36, 74	isso2i 5510	1 ⊢ <Q Or Q

Syntax hints:  ¬ wn 3   ↔ wb 208   ∧ wa 398   ∨ wo 843   ∧ w3a 1083   = wceq 1537   ∈ wcel 2114   class class class wbr 5068   Or wor 5475   × cxp 5555  ‘cfv 6357  (class class class)co 7158  1^st c1st 7689  2^nd c2nd 7690  Ncnpi 10268   ·_N cmi 10270   <_N clti 10271   ~_Q ceq 10275  Qcnq 10276   <_Q cltq 10282

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-sep 5205  ax-nul 5212  ax-pow 5268  ax-pr 5332  ax-un 7463

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ne 3019  df-ral 3145  df-rex 3146  df-reu 3147  df-rmo 3148  df-rab 3149  df-v 3498  df-sbc 3775  df-csb 3886  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-pss 3956  df-nul 4294  df-if 4470  df-pw 4543  df-sn 4570  df-pr 4572  df-tp 4574  df-op 4576  df-uni 4841  df-iun 4923  df-br 5069  df-opab 5131  df-mpt 5149  df-tr 5175  df-id 5462  df-eprel 5467  df-po 5476  df-so 5477  df-fr 5516  df-we 5518  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-rn 5568  df-res 5569  df-ima 5570  df-pred 6150  df-ord 6196  df-on 6197  df-lim 6198  df-suc 6199  df-iota 6316  df-fun 6359  df-fn 6360  df-f 6361  df-f1 6362  df-fo 6363  df-f1o 6364  df-fv 6365  df-ov 7161  df-oprab 7162  df-mpo 7163  df-om 7583  df-1st 7691  df-2nd 7692  df-wrecs 7949  df-recs 8010  df-rdg 8048  df-oadd 8108  df-omul 8109  df-er 8291  df-ni 10296  df-mi 10298  df-lti 10299  df-ltpq 10334  df-enq 10335  df-nq 10336  df-ltnq 10342

This theorem is referenced by:  ltbtwnnq  10402  prub  10418  npomex  10420  genpnnp  10429  nqpr  10438  distrlem4pr  10450  prlem934  10457  ltexprlem4  10463  reclem2pr  10472  reclem4pr  10474