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Theorem enq0enq 7515

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 7508	. . 3 ~_Q0 $/\$ $/\$ $/\$ $/\$
2		df-xp 4670	. . 3 $/\$
3	1, 2	ineq12i 3363	. 2 ~_Q0 $/\$ $/\$ $/\$ $/\$ $/\$
4		inopab 4799	. 2 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
5		an32 562	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
6		an4 586	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
7		pinn 7393	. . . . . . . . . . . . 13
8	7	ssriv 3188	. . . . . . . . . . . 12
9		xpss1 4774	. . . . . . . . . . . 12
10	8, 9	ax-mp 5	. . . . . . . . . . 11
11	10	sseli 3180	. . . . . . . . . 10
12	11	pm4.71ri 392	. . . . . . . . 9 $/\$
13	10	sseli 3180	. . . . . . . . . 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 2259	. . . . . . . . . . . . . . . . . . 19
20		opelxp 4694	. . . . . . . . . . . . . . . . . . 19 $/\$
21	19, 20	bitrdi 196	. . . . . . . . . . . . . . . . . 18 $/\$
22		eleq1 2259	. . . . . . . . . . . . . . . . . . 19
23		opelxp 4694	. . . . . . . . . . . . . . . . . . 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 7401	. . . . . . . . . . . . . . . . . 18 $/\$
31		mulpiord 7401	. . . . . . . . . . . . . . . . . 18 $/\$
32	30, 31	eqeqan12d 2212	. . . . . . . . . . . . . . . . 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 1620	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
49		19.42vv 1926	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
50		19.42vv 1926	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
51	48, 49, 50	3bitr3i 210	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
52	51	2exbii 1620	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
53		19.42vv 1926	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
54		19.42vv 1926	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
55	52, 53, 54	3bitr3i 210	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
56	18, 55	bitri 184	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
57	56	opabbii 4101	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
58		df-enq 7431	. . 3 $/\$ $/\$ $/\$ $/\$
59	57, 58	eqtr4i 2220	. 2 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
60	3, 4, 59	3eqtrri 2222	1 ~_Q0

Colors of variables: wff set class

Syntax hints: $/\$ wa 104

wb 105

wceq 1364

wex 1506

wcel 2167

cin 3156

wss 3157

cop 3626

copab 4094

com 4627

cxp 4662 (class class class)co 5925

comu 6481

cnpi 7356

cmi 7358

ceq 7363 ~_Q0 ceq0 7370

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 710 ax-5 1461 ax-7 1462 ax-gen 1463 ax-ie1 1507 ax-ie2 1508 ax-8 1518 ax-10 1519 ax-11 1520 ax-i12 1521 ax-bndl 1523 ax-4 1524 ax-17 1540 ax-i9 1544 ax-ial 1548 ax-i5r 1549 ax-14 2170 ax-ext 2178 ax-sep 4152 ax-pow 4208 ax-pr 4243

This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-nf 1475 df-sb 1777 df-clab 2183 df-cleq 2189 df-clel 2192 df-nfc 2328 df-ral 2480 df-rex 2481 df-v 2765 df-dif 3159 df-un 3161 df-in 3163 df-ss 3170 df-pw 3608 df-sn 3629 df-pr 3630 df-op 3632 df-uni 3841 df-br 4035 df-opab 4096 df-xp 4670 df-rel 4671 df-res 4676 df-iota 5220 df-fv 5267 df-ov 5928 df-ni 7388 df-mi 7390 df-enq 7431 df-enq0 7508

This theorem is referenced by: nqnq0pi 7522

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