elinp - Intuitionistic Logic Explorer

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

Theorem elinp 7415

Description: Membership in positive reals. (Contributed by Jim Kingdon, 27-Sep-2019.)

**Assertion**
Ref	Expression
elinp	$/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$

Distinct variable groups:

Proof of Theorem elinp Dummy variables

are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		npsspw 7412	. . . . 5
2	1	sseli 3138	. . . 4
3		opelxp 4634	. . . 4 $/\$
4	2, 3	sylib 121	. . 3 $/\$
5		elex 2737	. . . 4
6		elex 2737	. . . 4
7	5, 6	anim12i 336	. . 3 $/\$ $/\$
8	4, 7	syl 14	. 2 $/\$
9		nqex 7304	. . . . 5
10	9	ssex 4119	. . . 4
11	9	ssex 4119	. . . 4
12	10, 11	anim12i 336	. . 3 $/\$ $/\$
13	12	ad2antrr 480	. 2 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$ $/\$
14		df-inp 7407	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$
15	14	eleq2i 2233	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$
16		sseq1 3165	. . . . . . 7
17	16	anbi1d 461	. . . . . 6 $/\$ $/\$
18		eleq2 2230	. . . . . . . 8
19	18	rexbidv 2467	. . . . . . 7
20	19	anbi1d 461	. . . . . 6 $/\$ $/\$
21	17, 20	anbi12d 465	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
22		eleq2 2230	. . . . . . . . . . 11
23	22	anbi2d 460	. . . . . . . . . 10 $/\$ $/\$
24	23	rexbidv 2467	. . . . . . . . 9 $/\$ $/\$
25	18, 24	bibi12d 234	. . . . . . . 8 $/\$ $/\$
26	25	ralbidv 2466	. . . . . . 7 $/\$ $/\$
27	26	anbi1d 461	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
28	18	anbi1d 461	. . . . . . . 8 $/\$ $/\$
29	28	notbid 657	. . . . . . 7 $/\$ $/\$
30	29	ralbidv 2466	. . . . . 6 $/\$ $/\$
31	18	orbi1d 781	. . . . . . . 8 $\/$ $\/$
32	31	imbi2d 229	. . . . . . 7 $\/$ $\/$
33	32	2ralbidv 2490	. . . . . 6 $\/$ $\/$
34	27, 30, 33	3anbi123d 1302	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$
35	21, 34	anbi12d 465	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$
36		sseq1 3165	. . . . . . 7
37	36	anbi2d 460	. . . . . 6 $/\$ $/\$
38		eleq2 2230	. . . . . . . 8
39	38	rexbidv 2467	. . . . . . 7
40	39	anbi2d 460	. . . . . 6 $/\$ $/\$
41	37, 40	anbi12d 465	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
42		eleq2 2230	. . . . . . . . . . 11
43	42	anbi2d 460	. . . . . . . . . 10 $/\$ $/\$
44	43	rexbidv 2467	. . . . . . . . 9 $/\$ $/\$
45	38, 44	bibi12d 234	. . . . . . . 8 $/\$ $/\$
46	45	ralbidv 2466	. . . . . . 7 $/\$ $/\$
47	46	anbi2d 460	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
48	42	anbi2d 460	. . . . . . . 8 $/\$ $/\$
49	48	notbid 657	. . . . . . 7 $/\$ $/\$
50	49	ralbidv 2466	. . . . . 6 $/\$ $/\$
51	38	orbi2d 780	. . . . . . . 8 $\/$ $\/$
52	51	imbi2d 229	. . . . . . 7 $\/$ $\/$
53	52	2ralbidv 2490	. . . . . 6 $\/$ $\/$
54	47, 50, 53	3anbi123d 1302	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$
55	41, 54	anbi12d 465	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$
56	35, 55	opelopabg 4246	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$
57	15, 56	syl5bb 191	. 2 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$
58	8, 13, 57	pm5.21nii 694	1 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$

Colors of variables: wff set class

Syntax hints:

wn 3

wi 4 $/\$ wa 103

wb 104 $\/$ wo 698 $/\$ w3a 968

wceq 1343

wcel 2136

wral 2444

wrex 2445

cvv 2726

wss 3116

cpw 3559

cop 3579 class class class wbr 3982

copab 4042

cxp 4602

cnq 7221

cltq 7226

cnp 7232

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1435 ax-7 1436 ax-gen 1437 ax-ie1 1481 ax-ie2 1482 ax-8 1492 ax-10 1493 ax-11 1494 ax-i12 1495 ax-bndl 1497 ax-4 1498 ax-17 1514 ax-i9 1518 ax-ial 1522 ax-i5r 1523 ax-13 2138 ax-14 2139 ax-ext 2147 ax-coll 4097 ax-sep 4100 ax-pow 4153 ax-pr 4187 ax-un 4411 ax-iinf 4565

This theorem depends on definitions: df-bi 116 df-3an 970 df-tru 1346 df-nf 1449 df-sb 1751 df-eu 2017 df-mo 2018 df-clab 2152 df-cleq 2158 df-clel 2161 df-nfc 2297 df-ral 2449 df-rex 2450 df-reu 2451 df-rab 2453 df-v 2728 df-sbc 2952 df-csb 3046 df-dif 3118 df-un 3120 df-in 3122 df-ss 3129 df-pw 3561 df-sn 3582 df-pr 3583 df-op 3585 df-uni 3790 df-int 3825 df-iun 3868 df-br 3983 df-opab 4044 df-mpt 4045 df-id 4271 df-iom 4568 df-xp 4610 df-rel 4611 df-cnv 4612 df-co 4613 df-dm 4614 df-rn 4615 df-res 4616 df-ima 4617 df-iota 5153 df-fun 5190 df-fn 5191 df-f 5192 df-f1 5193 df-fo 5194 df-f1o 5195 df-fv 5196 df-qs 6507 df-ni 7245 df-nqqs 7289 df-inp 7407

This theorem is referenced by: elnp1st2nd 7417 prml 7418 prmu 7419 prssnql 7420 prssnqu 7421 prcdnql 7425 prcunqu 7426 prltlu 7428 prnmaxl 7429 prnminu 7430 prloc 7432 prdisj 7433 nqprxx 7487 suplocexprlemex 7663

W3C validator