enq0enq - Intuitionistic Logic Explorer

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

Theorem enq0enq 7711

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 7704	. . 3 ~_Q0 $/\$ $/\$ $/\$ $/\$
2		df-xp 4737	. . 3 $/\$
3	1, 2	ineq12i 3408	. 2 ~_Q0 $/\$ $/\$ $/\$ $/\$ $/\$
4		inopab 4868	. 2 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
5		an32 564	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
6		an4 588	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
7		pinn 7589	. . . . . . . . . . . . 13
8	7	ssriv 3232	. . . . . . . . . . . 12
9		xpss1 4842	. . . . . . . . . . . 12
10	8, 9	ax-mp 5	. . . . . . . . . . 11
11	10	sseli 3224	. . . . . . . . . 10
12	11	pm4.71ri 392	. . . . . . . . 9 $/\$
13	10	sseli 3224	. . . . . . . . . 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 2294	. . . . . . . . . . . . . . . . . . 19
20		opelxp 4761	. . . . . . . . . . . . . . . . . . 19 $/\$
21	19, 20	bitrdi 196	. . . . . . . . . . . . . . . . . 18 $/\$
22		eleq1 2294	. . . . . . . . . . . . . . . . . . 19
23		opelxp 4761	. . . . . . . . . . . . . . . . . . 19 $/\$
24	22, 23	bitrdi 196	. . . . . . . . . . . . . . . . . 18 $/\$
25	21, 24	bi2anan9 610	. . . . . . . . . . . . . . . . 17 $/\$ $/\$ $/\$ $/\$ $/\$
26	25	pm5.32i 454	. . . . . . . . . . . . . . . 16 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
27	26	anbi1i 458	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
28		anass 401	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
29	27, 28	bitri 184	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
30		mulpiord 7597	. . . . . . . . . . . . . . . . . 18 $/\$
31		mulpiord 7597	. . . . . . . . . . . . . . . . . 18 $/\$
32	30, 31	eqeqan12d 2247	. . . . . . . . . . . . . . . . 17 $/\$ $/\$ $/\$
33	32	an42s 593	. . . . . . . . . . . . . . . 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 1655	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
49		19.42vv 1960	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
50		19.42vv 1960	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
51	48, 49, 50	3bitr3i 210	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
52	51	2exbii 1655	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
53		19.42vv 1960	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
54		19.42vv 1960	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
55	52, 53, 54	3bitr3i 210	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
56	18, 55	bitri 184	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
57	56	opabbii 4161	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
58		df-enq 7627	. . 3 $/\$ $/\$ $/\$ $/\$
59	57, 58	eqtr4i 2255	. 2 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
60	3, 4, 59	3eqtrri 2257	1 ~_Q0

Colors of variables: wff set class

Syntax hints: $/\$ wa 104

wb 105

wceq 1398

wex 1541

wcel 2202

cin 3200

wss 3201

cop 3676

copab 4154

com 4694

cxp 4729 (class class class)co 6028

comu 6623

cnpi 7552

cmi 7554

ceq 7559 ~_Q0 ceq0 7566

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 619 ax-in2 620 ax-io 717 ax-5 1496 ax-7 1497 ax-gen 1498 ax-ie1 1542 ax-ie2 1543 ax-8 1553 ax-10 1554 ax-11 1555 ax-i12 1556 ax-bndl 1558 ax-4 1559 ax-17 1575 ax-i9 1579 ax-ial 1583 ax-i5r 1584 ax-14 2205 ax-ext 2213 ax-sep 4212 ax-pow 4270 ax-pr 4305

This theorem depends on definitions: df-bi 117 df-3an 1007 df-tru 1401 df-nf 1510 df-sb 1811 df-clab 2218 df-cleq 2224 df-clel 2227 df-nfc 2364 df-ral 2516 df-rex 2517 df-v 2805 df-dif 3203 df-un 3205 df-in 3207 df-ss 3214 df-pw 3658 df-sn 3679 df-pr 3680 df-op 3682 df-uni 3899 df-br 4094 df-opab 4156 df-xp 4737 df-rel 4738 df-res 4743 df-iota 5293 df-fv 5341 df-ov 6031 df-ni 7584 df-mi 7586 df-enq 7627 df-enq0 7704

This theorem is referenced by: nqnq0pi 7718

W3C validator