Theorem ltsonq 10870

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 10826	. . . . . . 7 ⊢ (𝑥 ∈ Q → 𝑥 ∈ (N × N))
2	1	adantr 480	. . . . . 6 ⊢ ((𝑥 ∈ Q ∧ 𝑦 ∈ Q) → 𝑥 ∈ (N × N))
3		xp1st 7962	. . . . . 6 ⊢ (𝑥 ∈ (N × N) → (1st ‘𝑥) ∈ N)
4	2, 3	syl 17	. . . . 5 ⊢ ((𝑥 ∈ Q ∧ 𝑦 ∈ Q) → (1st ‘𝑥) ∈ N)
5		elpqn 10826	. . . . . . 7 ⊢ (𝑦 ∈ Q → 𝑦 ∈ (N × N))
6	5	adantl 481	. . . . . 6 ⊢ ((𝑥 ∈ Q ∧ 𝑦 ∈ Q) → 𝑦 ∈ (N × N))
7		xp2nd 7963	. . . . . 6 ⊢ (𝑦 ∈ (N × N) → (2nd ‘𝑦) ∈ N)
8	6, 7	syl 17	. . . . 5 ⊢ ((𝑥 ∈ Q ∧ 𝑦 ∈ Q) → (2nd ‘𝑦) ∈ N)
9		mulclpi 10794	. . . . 5 ⊢ (((1st ‘𝑥) ∈ N ∧ (2nd ‘𝑦) ∈ N) → ((1st ‘𝑥) ·N (2nd ‘𝑦)) ∈ N)
10	4, 8, 9	syl2anc 584	. . . 4 ⊢ ((𝑥 ∈ Q ∧ 𝑦 ∈ Q) → ((1st ‘𝑥) ·N (2nd ‘𝑦)) ∈ N)
11		xp1st 7962	. . . . . 6 ⊢ (𝑦 ∈ (N × N) → (1st ‘𝑦) ∈ N)
12	6, 11	syl 17	. . . . 5 ⊢ ((𝑥 ∈ Q ∧ 𝑦 ∈ Q) → (1st ‘𝑦) ∈ N)
13		xp2nd 7963	. . . . . 6 ⊢ (𝑥 ∈ (N × N) → (2nd ‘𝑥) ∈ N)
14	2, 13	syl 17	. . . . 5 ⊢ ((𝑥 ∈ Q ∧ 𝑦 ∈ Q) → (2nd ‘𝑥) ∈ N)
15		mulclpi 10794	. . . . 5 ⊢ (((1st ‘𝑦) ∈ N ∧ (2nd ‘𝑥) ∈ N) → ((1st ‘𝑦) ·N (2nd ‘𝑥)) ∈ N)
16	12, 14, 15	syl2anc 584	. . . 4 ⊢ ((𝑥 ∈ Q ∧ 𝑦 ∈ Q) → ((1st ‘𝑦) ·N (2nd ‘𝑥)) ∈ N)
17		ltsopi 10789	. . . . 5 ⊢ <N Or N
18		sotric 5559	. . . . 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 690	. . . 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 584	. . 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 10844	. . 3 ⊢ ((𝑥 ∈ Q ∧ 𝑦 ∈ Q) → (𝑥 <Q 𝑦 ↔ ((1st ‘𝑥) ·N (2nd ‘𝑦)) <N ((1st ‘𝑦) ·N (2nd ‘𝑥))))
22		fveq2 6831	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (1st ‘𝑥) = (1st ‘𝑦))
23		fveq2 6831	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (2nd ‘𝑥) = (2nd ‘𝑦))
24	23	eqcomd 2739	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (2nd ‘𝑦) = (2nd ‘𝑥))
25	22, 24	oveq12d 7373	. . . . . 6 ⊢ (𝑥 = 𝑦 → ((1st ‘𝑥) ·N (2nd ‘𝑦)) = ((1st ‘𝑦) ·N (2nd ‘𝑥)))
26		enqbreq2 10821	. . . . . . . 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 596	. . . . . . 7 ⊢ ((𝑥 ∈ Q ∧ 𝑦 ∈ Q) → (𝑥 ~Q 𝑦 ↔ ((1st ‘𝑥) ·N (2nd ‘𝑦)) = ((1st ‘𝑦) ·N (2nd ‘𝑥))))
28		enqeq 10835	. . . . . . . 8 ⊢ ((𝑥 ∈ Q ∧ 𝑦 ∈ Q ∧ 𝑥 ~Q 𝑦) → 𝑥 = 𝑦)
29	28	3expia 1121	. . . . . . 7 ⊢ ((𝑥 ∈ Q ∧ 𝑦 ∈ Q) → (𝑥 ~Q 𝑦 → 𝑥 = 𝑦))
30	27, 29	sylbird 260	. . . . . 6 ⊢ ((𝑥 ∈ Q ∧ 𝑦 ∈ Q) → (((1st ‘𝑥) ·N (2nd ‘𝑦)) = ((1st ‘𝑦) ·N (2nd ‘𝑥)) → 𝑥 = 𝑦))
31	25, 30	impbid2 226	. . . . 5 ⊢ ((𝑥 ∈ Q ∧ 𝑦 ∈ Q) → (𝑥 = 𝑦 ↔ ((1st ‘𝑥) ·N (2nd ‘𝑦)) = ((1st ‘𝑦) ·N (2nd ‘𝑥))))
32		ordpinq 10844	. . . . . 6 ⊢ ((𝑦 ∈ Q ∧ 𝑥 ∈ Q) → (𝑦 <Q 𝑥 ↔ ((1st ‘𝑦) ·N (2nd ‘𝑥)) <N ((1st ‘𝑥) ·N (2nd ‘𝑦))))
33	32	ancoms 458	. . . . 5 ⊢ ((𝑥 ∈ Q ∧ 𝑦 ∈ Q) → (𝑦 <Q 𝑥 ↔ ((1st ‘𝑦) ·N (2nd ‘𝑥)) <N ((1st ‘𝑥) ·N (2nd ‘𝑦))))
34	31, 33	orbi12d 918	. . . 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 318	. . 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 311	. 2 ⊢ ((𝑥 ∈ Q ∧ 𝑦 ∈ Q) → (𝑥 <Q 𝑦 ↔ ¬ (𝑥 = 𝑦 ∨ 𝑦 <Q 𝑥)))
37	21	3adant3 1132	. . . . . 6 ⊢ ((𝑥 ∈ Q ∧ 𝑦 ∈ Q ∧ 𝑧 ∈ Q) → (𝑥 <Q 𝑦 ↔ ((1st ‘𝑥) ·N (2nd ‘𝑦)) <N ((1st ‘𝑦) ·N (2nd ‘𝑥))))
38		elpqn 10826	. . . . . . . 8 ⊢ (𝑧 ∈ Q → 𝑧 ∈ (N × N))
39	38	3ad2ant3 1135	. . . . . . 7 ⊢ ((𝑥 ∈ Q ∧ 𝑦 ∈ Q ∧ 𝑧 ∈ Q) → 𝑧 ∈ (N × N))
40		xp2nd 7963	. . . . . . 7 ⊢ (𝑧 ∈ (N × N) → (2nd ‘𝑧) ∈ N)
41		ltmpi 10805	. . . . . . 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 279	. . . . 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 10844	. . . . . . 7 ⊢ ((𝑦 ∈ Q ∧ 𝑧 ∈ Q) → (𝑦 <Q 𝑧 ↔ ((1st ‘𝑦) ·N (2nd ‘𝑧)) <N ((1st ‘𝑧) ·N (2nd ‘𝑦))))
45	44	3adant1 1130	. . . . . 6 ⊢ ((𝑥 ∈ Q ∧ 𝑦 ∈ Q ∧ 𝑧 ∈ Q) → (𝑦 <Q 𝑧 ↔ ((1st ‘𝑦) ·N (2nd ‘𝑧)) <N ((1st ‘𝑧) ·N (2nd ‘𝑦))))
46	1	3ad2ant1 1133	. . . . . . 7 ⊢ ((𝑥 ∈ Q ∧ 𝑦 ∈ Q ∧ 𝑧 ∈ Q) → 𝑥 ∈ (N × N))
47		ltmpi 10805	. . . . . . 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 279	. . . . 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 6844	. . . . . . 7 ⊢ (2nd ‘𝑥) ∈ V
52		fvex 6844	. . . . . . 7 ⊢ (1st ‘𝑦) ∈ V
53		fvex 6844	. . . . . . 7 ⊢ (2nd ‘𝑧) ∈ V
54		mulcompi 10797	. . . . . . 7 ⊢ (𝑟 ·N 𝑠) = (𝑠 ·N 𝑟)
55		mulasspi 10798	. . . . . . 7 ⊢ ((𝑟 ·N 𝑠) ·N 𝑡) = (𝑟 ·N (𝑠 ·N 𝑡))
56	51, 52, 53, 54, 55	caov13 7585	. . . . . 6 ⊢ ((2nd ‘𝑥) ·N ((1st ‘𝑦) ·N (2nd ‘𝑧))) = ((2nd ‘𝑧) ·N ((1st ‘𝑦) ·N (2nd ‘𝑥)))
57		fvex 6844	. . . . . . 7 ⊢ (1st ‘𝑧) ∈ V
58		fvex 6844	. . . . . . 7 ⊢ (2nd ‘𝑦) ∈ V
59	51, 57, 58, 54, 55	caov13 7585	. . . . . 6 ⊢ ((2nd ‘𝑥) ·N ((1st ‘𝑧) ·N (2nd ‘𝑦))) = ((2nd ‘𝑦) ·N ((1st ‘𝑧) ·N (2nd ‘𝑥)))
60	56, 59	breq12i 5104	. . . . 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 6844	. . . . . . 7 ⊢ (1st ‘𝑥) ∈ V
62	53, 61, 58, 54, 55	caov13 7585	. . . . . 6 ⊢ ((2nd ‘𝑧) ·N ((1st ‘𝑥) ·N (2nd ‘𝑦))) = ((2nd ‘𝑦) ·N ((1st ‘𝑥) ·N (2nd ‘𝑧)))
63		ltrelpi 10790	. . . . . . 7 ⊢ <N ⊆ (N × N)
64	17, 63	sotri 6081	. . . . . 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 5131	. . . . 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 594	. . . 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 253	. . 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 10844	. . . . 5 ⊢ ((𝑥 ∈ Q ∧ 𝑧 ∈ Q) → (𝑥 <Q 𝑧 ↔ ((1st ‘𝑥) ·N (2nd ‘𝑧)) <N ((1st ‘𝑧) ·N (2nd ‘𝑥))))
69	68	3adant2 1131	. . . 4 ⊢ ((𝑥 ∈ Q ∧ 𝑦 ∈ Q ∧ 𝑧 ∈ Q) → (𝑥 <Q 𝑧 ↔ ((1st ‘𝑥) ·N (2nd ‘𝑧)) <N ((1st ‘𝑧) ·N (2nd ‘𝑥))))
70	5	3ad2ant2 1134	. . . . 5 ⊢ ((𝑥 ∈ Q ∧ 𝑦 ∈ Q ∧ 𝑧 ∈ Q) → 𝑦 ∈ (N × N))
71		ltmpi 10805	. . . . 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 279	. . 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 259	. 2 ⊢ ((𝑥 ∈ Q ∧ 𝑦 ∈ Q ∧ 𝑧 ∈ Q) → ((𝑥 <Q 𝑦 ∧ 𝑦 <Q 𝑧) → 𝑥 <Q 𝑧))
75	36, 74	isso2i 5566	1 ⊢ <Q Or Q

Syntax hints:  ¬ wn 3   ↔ wb 206   ∧ wa 395   ∨ wo 847   ∧ w3a 1086   = wceq 1541   ∈ wcel 2113   class class class wbr 5095   Or wor 5528   × cxp 5619  ‘cfv 6489  (class class class)co 7355  1^st c1st 7928  2^nd c2nd 7929  Ncnpi 10745   ·_N cmi 10747   <_N clti 10748   ~_Q ceq 10752  Qcnq 10753   <_Q cltq 10759

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 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-sep 5238  ax-nul 5248  ax-pr 5374  ax-un 7677

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2883  df-ne 2931  df-ral 3050  df-rex 3059  df-rmo 3348  df-reu 3349  df-rab 3398  df-v 3440  df-sbc 3739  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4861  df-iun 4945  df-br 5096  df-opab 5158  df-mpt 5177  df-tr 5203  df-id 5516  df-eprel 5521  df-po 5529  df-so 5530  df-fr 5574  df-we 5576  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-pred 6256  df-ord 6317  df-on 6318  df-lim 6319  df-suc 6320  df-iota 6445  df-fun 6491  df-fn 6492  df-f 6493  df-f1 6494  df-fo 6495  df-f1o 6496  df-fv 6497  df-ov 7358  df-oprab 7359  df-mpo 7360  df-om 7806  df-1st 7930  df-2nd 7931  df-frecs 8220  df-wrecs 8251  df-recs 8300  df-rdg 8338  df-oadd 8398  df-omul 8399  df-er 8631  df-ni 10773  df-mi 10775  df-lti 10776  df-ltpq 10811  df-enq 10812  df-nq 10813  df-ltnq 10819

This theorem is referenced by:  ltbtwnnq  10879  prub  10895  npomex  10897  genpnnp  10906  nqpr  10915  distrlem4pr  10927  prlem934  10934  ltexprlem4  10940  reclem2pr  10949  reclem4pr  10951