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Theorem enq0enq 7544

Description: Equivalence on positive fractions in terms of equivalence on nonnegative fractions. (Contributed by Jim Kingdon, 12-Nov-2019.)

**Assertion**
Ref	Expression
enq0enq	~_Q0

Proof of Theorem enq0enq Dummy variables

are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-enq0 7537	. . 3 ~_Q0 $/\$ $/\$ $/\$ $/\$
2		df-xp 4681	. . 3 $/\$
3	1, 2	ineq12i 3372	. 2 ~_Q0 $/\$ $/\$ $/\$ $/\$ $/\$
4		inopab 4810	. 2 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
5		an32 562	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
6		an4 586	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
7		pinn 7422	. . . . . . . . . . . . 13
8	7	ssriv 3197	. . . . . . . . . . . 12
9		xpss1 4785	. . . . . . . . . . . 12
10	8, 9	ax-mp 5	. . . . . . . . . . 11
11	10	sseli 3189	. . . . . . . . . 10
12	11	pm4.71ri 392	. . . . . . . . 9 $/\$
13	10	sseli 3189	. . . . . . . . . 10
14	13	pm4.71ri 392	. . . . . . . . 9 $/\$
15	12, 14	anbi12i 460	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
16	6, 15	bitr4i 187	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
17	16	anbi1i 458	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
18	5, 17	bitri 184	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
19		eleq1 2268	. . . . . . . . . . . . . . . . . . 19
20		opelxp 4705	. . . . . . . . . . . . . . . . . . 19 $/\$
21	19, 20	bitrdi 196	. . . . . . . . . . . . . . . . . 18 $/\$
22		eleq1 2268	. . . . . . . . . . . . . . . . . . 19
23		opelxp 4705	. . . . . . . . . . . . . . . . . . 19 $/\$
24	22, 23	bitrdi 196	. . . . . . . . . . . . . . . . . 18 $/\$
25	21, 24	bi2anan9 606	. . . . . . . . . . . . . . . . 17 $/\$ $/\$ $/\$ $/\$ $/\$
26	25	pm5.32i 454	. . . . . . . . . . . . . . . 16 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
27	26	anbi1i 458	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
28		anass 401	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
29	27, 28	bitri 184	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
30		mulpiord 7430	. . . . . . . . . . . . . . . . . 18 $/\$
31		mulpiord 7430	. . . . . . . . . . . . . . . . . 18 $/\$
32	30, 31	eqeqan12d 2221	. . . . . . . . . . . . . . . . 17 $/\$ $/\$ $/\$
33	32	an42s 589	. . . . . . . . . . . . . . . 16 $/\$ $/\$ $/\$
34	33	pm5.32i 454	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
35	34	anbi2i 457	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
36	29, 35	bitr4i 187	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
37		anass 401	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
38	36, 37	bitr4i 187	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
39	26	anbi1i 458	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
40	38, 39	bitr4i 187	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
41		ancom 266	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
42	41	anbi1i 458	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
43	41	anbi1i 458	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
44	40, 42, 43	3bitr3i 210	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
45		anass 401	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
46		anass 401	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
47	44, 45, 46	3bitr3i 210	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
48	47	2exbii 1629	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
49		19.42vv 1935	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
50		19.42vv 1935	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
51	48, 49, 50	3bitr3i 210	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
52	51	2exbii 1629	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
53		19.42vv 1935	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
54		19.42vv 1935	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
55	52, 53, 54	3bitr3i 210	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
56	18, 55	bitri 184	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
57	56	opabbii 4111	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
58		df-enq 7460	. . 3 $/\$ $/\$ $/\$ $/\$
59	57, 58	eqtr4i 2229	. 2 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
60	3, 4, 59	3eqtrri 2231	1 ~_Q0

Colors of variables: wff set class

Syntax hints: $/\$ wa 104

wb 105

wceq 1373

wex 1515

wcel 2176

cin 3165

wss 3166

cop 3636

copab 4104

com 4638

cxp 4673 (class class class)co 5944

comu 6500

cnpi 7385

cmi 7387

ceq 7392 ~_Q0 ceq0 7399

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 711 ax-5 1470 ax-7 1471 ax-gen 1472 ax-ie1 1516 ax-ie2 1517 ax-8 1527 ax-10 1528 ax-11 1529 ax-i12 1530 ax-bndl 1532 ax-4 1533 ax-17 1549 ax-i9 1553 ax-ial 1557 ax-i5r 1558 ax-14 2179 ax-ext 2187 ax-sep 4162 ax-pow 4218 ax-pr 4253

This theorem depends on definitions: df-bi 117 df-3an 983 df-tru 1376 df-nf 1484 df-sb 1786 df-clab 2192 df-cleq 2198 df-clel 2201 df-nfc 2337 df-ral 2489 df-rex 2490 df-v 2774 df-dif 3168 df-un 3170 df-in 3172 df-ss 3179 df-pw 3618 df-sn 3639 df-pr 3640 df-op 3642 df-uni 3851 df-br 4045 df-opab 4106 df-xp 4681 df-rel 4682 df-res 4687 df-iota 5232 df-fv 5279 df-ov 5947 df-ni 7417 df-mi 7419 df-enq 7460 df-enq0 7537

This theorem is referenced by: nqnq0pi 7551

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