Theorem ltsonq 10929

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 10885	. . . . . . 7 ⊢ (𝑥 ∈ Q → 𝑥 ∈ (N × N))
2	1	adantr 480	. . . . . 6 ⊢ ((𝑥 ∈ Q ∧ 𝑦 ∈ Q) → 𝑥 ∈ (N × N))
3		xp1st 8003	. . . . . 6 ⊢ (𝑥 ∈ (N × N) → (1st ‘𝑥) ∈ N)
4	2, 3	syl 17	. . . . 5 ⊢ ((𝑥 ∈ Q ∧ 𝑦 ∈ Q) → (1st ‘𝑥) ∈ N)
5		elpqn 10885	. . . . . . 7 ⊢ (𝑦 ∈ Q → 𝑦 ∈ (N × N))
6	5	adantl 481	. . . . . 6 ⊢ ((𝑥 ∈ Q ∧ 𝑦 ∈ Q) → 𝑦 ∈ (N × N))
7		xp2nd 8004	. . . . . 6 ⊢ (𝑦 ∈ (N × N) → (2nd ‘𝑦) ∈ N)
8	6, 7	syl 17	. . . . 5 ⊢ ((𝑥 ∈ Q ∧ 𝑦 ∈ Q) → (2nd ‘𝑦) ∈ N)
9		mulclpi 10853	. . . . 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 8003	. . . . . 6 ⊢ (𝑦 ∈ (N × N) → (1st ‘𝑦) ∈ N)
12	6, 11	syl 17	. . . . 5 ⊢ ((𝑥 ∈ Q ∧ 𝑦 ∈ Q) → (1st ‘𝑦) ∈ N)
13		xp2nd 8004	. . . . . 6 ⊢ (𝑥 ∈ (N × N) → (2nd ‘𝑥) ∈ N)
14	2, 13	syl 17	. . . . 5 ⊢ ((𝑥 ∈ Q ∧ 𝑦 ∈ Q) → (2nd ‘𝑥) ∈ N)
15		mulclpi 10853	. . . . 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 10848	. . . . 5 ⊢ <N Or N
18		sotric 5579	. . . . 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 10903	. . 3 ⊢ ((𝑥 ∈ Q ∧ 𝑦 ∈ Q) → (𝑥 <Q 𝑦 ↔ ((1st ‘𝑥) ·N (2nd ‘𝑦)) <N ((1st ‘𝑦) ·N (2nd ‘𝑥))))
22		fveq2 6861	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (1st ‘𝑥) = (1st ‘𝑦))
23		fveq2 6861	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (2nd ‘𝑥) = (2nd ‘𝑦))
24	23	eqcomd 2736	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (2nd ‘𝑦) = (2nd ‘𝑥))
25	22, 24	oveq12d 7408	. . . . . 6 ⊢ (𝑥 = 𝑦 → ((1st ‘𝑥) ·N (2nd ‘𝑦)) = ((1st ‘𝑦) ·N (2nd ‘𝑥)))
26		enqbreq2 10880	. . . . . . . 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 10894	. . . . . . . 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 10903	. . . . . 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 10885	. . . . . . . 8 ⊢ (𝑧 ∈ Q → 𝑧 ∈ (N × N))
39	38	3ad2ant3 1135	. . . . . . 7 ⊢ ((𝑥 ∈ Q ∧ 𝑦 ∈ Q ∧ 𝑧 ∈ Q) → 𝑧 ∈ (N × N))
40		xp2nd 8004	. . . . . . 7 ⊢ (𝑧 ∈ (N × N) → (2nd ‘𝑧) ∈ N)
41		ltmpi 10864	. . . . . . 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 10903	. . . . . . 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 10864	. . . . . . 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 6874	. . . . . . 7 ⊢ (2nd ‘𝑥) ∈ V
52		fvex 6874	. . . . . . 7 ⊢ (1st ‘𝑦) ∈ V
53		fvex 6874	. . . . . . 7 ⊢ (2nd ‘𝑧) ∈ V
54		mulcompi 10856	. . . . . . 7 ⊢ (𝑟 ·N 𝑠) = (𝑠 ·N 𝑟)
55		mulasspi 10857	. . . . . . 7 ⊢ ((𝑟 ·N 𝑠) ·N 𝑡) = (𝑟 ·N (𝑠 ·N 𝑡))
56	51, 52, 53, 54, 55	caov13 7622	. . . . . 6 ⊢ ((2nd ‘𝑥) ·N ((1st ‘𝑦) ·N (2nd ‘𝑧))) = ((2nd ‘𝑧) ·N ((1st ‘𝑦) ·N (2nd ‘𝑥)))
57		fvex 6874	. . . . . . 7 ⊢ (1st ‘𝑧) ∈ V
58		fvex 6874	. . . . . . 7 ⊢ (2nd ‘𝑦) ∈ V
59	51, 57, 58, 54, 55	caov13 7622	. . . . . 6 ⊢ ((2nd ‘𝑥) ·N ((1st ‘𝑧) ·N (2nd ‘𝑦))) = ((2nd ‘𝑦) ·N ((1st ‘𝑧) ·N (2nd ‘𝑥)))
60	56, 59	breq12i 5119	. . . . 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 6874	. . . . . . 7 ⊢ (1st ‘𝑥) ∈ V
62	53, 61, 58, 54, 55	caov13 7622	. . . . . 6 ⊢ ((2nd ‘𝑧) ·N ((1st ‘𝑥) ·N (2nd ‘𝑦))) = ((2nd ‘𝑦) ·N ((1st ‘𝑥) ·N (2nd ‘𝑧)))
63		ltrelpi 10849	. . . . . . 7 ⊢ <N ⊆ (N × N)
64	17, 63	sotri 6103	. . . . . 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 5146	. . . . 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 10903	. . . . 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 10864	. . . . 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 5586	1 ⊢ <Q Or Q

Syntax hints:  ¬ wn 3   ↔ wb 206   ∧ wa 395   ∨ wo 847   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109   class class class wbr 5110   Or wor 5548   × cxp 5639  ‘cfv 6514  (class class class)co 7390  1^st c1st 7969  2^nd c2nd 7970  Ncnpi 10804   ·_N cmi 10806   <_N clti 10807   ~_Q ceq 10811  Qcnq 10812   <_Q cltq 10818

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-sep 5254  ax-nul 5264  ax-pr 5390  ax-un 7714

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-rmo 3356  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-pss 3937  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-iun 4960  df-br 5111  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5536  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5594  df-we 5596  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-pred 6277  df-ord 6338  df-on 6339  df-lim 6340  df-suc 6341  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-ov 7393  df-oprab 7394  df-mpo 7395  df-om 7846  df-1st 7971  df-2nd 7972  df-frecs 8263  df-wrecs 8294  df-recs 8343  df-rdg 8381  df-oadd 8441  df-omul 8442  df-er 8674  df-ni 10832  df-mi 10834  df-lti 10835  df-ltpq 10870  df-enq 10871  df-nq 10872  df-ltnq 10878

This theorem is referenced by:  ltbtwnnq  10938  prub  10954  npomex  10956  genpnnp  10965  nqpr  10974  distrlem4pr  10986  prlem934  10993  ltexprlem4  10999  reclem2pr  11008  reclem4pr  11010