enq0enq - Intuitionistic Logic Explorer

	Intuitionistic Logic Explorer		< Previous Next > Nearby theorems
Mirrors > Home > ILE Home > Th. List > enq0enq		Unicode version

Theorem enq0enq 7618

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 7611	. . 3 ~_Q0 $/\$ $/\$ $/\$ $/\$
2		df-xp 4725	. . 3 $/\$
3	1, 2	ineq12i 3403	. 2 ~_Q0 $/\$ $/\$ $/\$ $/\$ $/\$
4		inopab 4854	. 2 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
5		an32 562	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
6		an4 586	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
7		pinn 7496	. . . . . . . . . . . . 13
8	7	ssriv 3228	. . . . . . . . . . . 12
9		xpss1 4829	. . . . . . . . . . . 12
10	8, 9	ax-mp 5	. . . . . . . . . . 11
11	10	sseli 3220	. . . . . . . . . 10
12	11	pm4.71ri 392	. . . . . . . . 9 $/\$
13	10	sseli 3220	. . . . . . . . . 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 2292	. . . . . . . . . . . . . . . . . . 19
20		opelxp 4749	. . . . . . . . . . . . . . . . . . 19 $/\$
21	19, 20	bitrdi 196	. . . . . . . . . . . . . . . . . 18 $/\$
22		eleq1 2292	. . . . . . . . . . . . . . . . . . 19
23		opelxp 4749	. . . . . . . . . . . . . . . . . . 19 $/\$
24	22, 23	bitrdi 196	. . . . . . . . . . . . . . . . . 18 $/\$
25	21, 24	bi2anan9 608	. . . . . . . . . . . . . . . . 17 $/\$ $/\$ $/\$ $/\$ $/\$
26	25	pm5.32i 454	. . . . . . . . . . . . . . . 16 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
27	26	anbi1i 458	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
28		anass 401	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
29	27, 28	bitri 184	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
30		mulpiord 7504	. . . . . . . . . . . . . . . . . 18 $/\$
31		mulpiord 7504	. . . . . . . . . . . . . . . . . 18 $/\$
32	30, 31	eqeqan12d 2245	. . . . . . . . . . . . . . . . 17 $/\$ $/\$ $/\$
33	32	an42s 591	. . . . . . . . . . . . . . . 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 1652	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
49		19.42vv 1958	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
50		19.42vv 1958	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
51	48, 49, 50	3bitr3i 210	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
52	51	2exbii 1652	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
53		19.42vv 1958	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
54		19.42vv 1958	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
55	52, 53, 54	3bitr3i 210	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
56	18, 55	bitri 184	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
57	56	opabbii 4151	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
58		df-enq 7534	. . 3 $/\$ $/\$ $/\$ $/\$
59	57, 58	eqtr4i 2253	. 2 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
60	3, 4, 59	3eqtrri 2255	1 ~_Q0

Colors of variables: wff set class

Syntax hints: $/\$ wa 104

wb 105

wceq 1395

wex 1538

wcel 2200

cin 3196

wss 3197

cop 3669

copab 4144

com 4682

cxp 4717 (class class class)co 6001

comu 6560

cnpi 7459

cmi 7461

ceq 7466 ~_Q0 ceq0 7473

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 617 ax-in2 618 ax-io 714 ax-5 1493 ax-7 1494 ax-gen 1495 ax-ie1 1539 ax-ie2 1540 ax-8 1550 ax-10 1551 ax-11 1552 ax-i12 1553 ax-bndl 1555 ax-4 1556 ax-17 1572 ax-i9 1576 ax-ial 1580 ax-i5r 1581 ax-14 2203 ax-ext 2211 ax-sep 4202 ax-pow 4258 ax-pr 4293

This theorem depends on definitions: df-bi 117 df-3an 1004 df-tru 1398 df-nf 1507 df-sb 1809 df-clab 2216 df-cleq 2222 df-clel 2225 df-nfc 2361 df-ral 2513 df-rex 2514 df-v 2801 df-dif 3199 df-un 3201 df-in 3203 df-ss 3210 df-pw 3651 df-sn 3672 df-pr 3673 df-op 3675 df-uni 3889 df-br 4084 df-opab 4146 df-xp 4725 df-rel 4726 df-res 4731 df-iota 5278 df-fv 5326 df-ov 6004 df-ni 7491 df-mi 7493 df-enq 7534 df-enq0 7611

This theorem is referenced by: nqnq0pi 7625

W3C validator