elinp - Intuitionistic Logic Explorer

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

Theorem elinp 7306

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 7303	. . . . 5
2	1	sseli 3098	. . . 4
3		opelxp 4577	. . . 4 $/\$
4	2, 3	sylib 121	. . 3 $/\$
5		elex 2700	. . . 4
6		elex 2700	. . . 4
7	5, 6	anim12i 336	. . 3 $/\$ $/\$
8	4, 7	syl 14	. 2 $/\$
9		nqex 7195	. . . . 5
10	9	ssex 4073	. . . 4
11	9	ssex 4073	. . . 4
12	10, 11	anim12i 336	. . 3 $/\$ $/\$
13	12	ad2antrr 480	. 2 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$ $/\$
14		df-inp 7298	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$
15	14	eleq2i 2207	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$
16		sseq1 3125	. . . . . . 7
17	16	anbi1d 461	. . . . . 6 $/\$ $/\$
18		eleq2 2204	. . . . . . . 8
19	18	rexbidv 2439	. . . . . . 7
20	19	anbi1d 461	. . . . . 6 $/\$ $/\$
21	17, 20	anbi12d 465	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
22		eleq2 2204	. . . . . . . . . . 11
23	22	anbi2d 460	. . . . . . . . . 10 $/\$ $/\$
24	23	rexbidv 2439	. . . . . . . . 9 $/\$ $/\$
25	18, 24	bibi12d 234	. . . . . . . 8 $/\$ $/\$
26	25	ralbidv 2438	. . . . . . 7 $/\$ $/\$
27	26	anbi1d 461	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
28	18	anbi1d 461	. . . . . . . 8 $/\$ $/\$
29	28	notbid 657	. . . . . . 7 $/\$ $/\$
30	29	ralbidv 2438	. . . . . 6 $/\$ $/\$
31	18	orbi1d 781	. . . . . . . 8 $\/$ $\/$
32	31	imbi2d 229	. . . . . . 7 $\/$ $\/$
33	32	2ralbidv 2462	. . . . . 6 $\/$ $\/$
34	27, 30, 33	3anbi123d 1291	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$
35	21, 34	anbi12d 465	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$
36		sseq1 3125	. . . . . . 7
37	36	anbi2d 460	. . . . . 6 $/\$ $/\$
38		eleq2 2204	. . . . . . . 8
39	38	rexbidv 2439	. . . . . . 7
40	39	anbi2d 460	. . . . . 6 $/\$ $/\$
41	37, 40	anbi12d 465	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
42		eleq2 2204	. . . . . . . . . . 11
43	42	anbi2d 460	. . . . . . . . . 10 $/\$ $/\$
44	43	rexbidv 2439	. . . . . . . . 9 $/\$ $/\$
45	38, 44	bibi12d 234	. . . . . . . 8 $/\$ $/\$
46	45	ralbidv 2438	. . . . . . 7 $/\$ $/\$
47	46	anbi2d 460	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
48	42	anbi2d 460	. . . . . . . 8 $/\$ $/\$
49	48	notbid 657	. . . . . . 7 $/\$ $/\$
50	49	ralbidv 2438	. . . . . 6 $/\$ $/\$
51	38	orbi2d 780	. . . . . . . 8 $\/$ $\/$
52	51	imbi2d 229	. . . . . . 7 $\/$ $\/$
53	52	2ralbidv 2462	. . . . . 6 $\/$ $\/$
54	47, 50, 53	3anbi123d 1291	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$
55	41, 54	anbi12d 465	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$
56	35, 55	opelopabg 4198	. . 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 963

wceq 1332

wcel 1481

wral 2417

wrex 2418

cvv 2689

wss 3076

cpw 3515

cop 3535 class class class wbr 3937

copab 3996

cxp 4545

cnq 7112

cltq 7117

cnp 7123

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 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-13 1492 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-coll 4051 ax-sep 4054 ax-pow 4106 ax-pr 4139 ax-un 4363 ax-iinf 4510

This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ral 2422 df-rex 2423 df-reu 2424 df-rab 2426 df-v 2691 df-sbc 2914 df-csb 3008 df-dif 3078 df-un 3080 df-in 3082 df-ss 3089 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-int 3780 df-iun 3823 df-br 3938 df-opab 3998 df-mpt 3999 df-id 4223 df-iom 4513 df-xp 4553 df-rel 4554 df-cnv 4555 df-co 4556 df-dm 4557 df-rn 4558 df-res 4559 df-ima 4560 df-iota 5096 df-fun 5133 df-fn 5134 df-f 5135 df-f1 5136 df-fo 5137 df-f1o 5138 df-fv 5139 df-qs 6443 df-ni 7136 df-nqqs 7180 df-inp 7298

This theorem is referenced by: elnp1st2nd 7308 prml 7309 prmu 7310 prssnql 7311 prssnqu 7312 prcdnql 7316 prcunqu 7317 prltlu 7319 prnmaxl 7320 prnminu 7321 prloc 7323 prdisj 7324 nqprxx 7378 suplocexprlemex 7554

W3C validator