elinp - Intuitionistic Logic Explorer

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Theorem elinp 6572

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 6569	. . . . 5
2	1	sseli 2941	. . . 4
3		opelxp 4374	. . . 4 $/\$
4	2, 3	sylib 127	. . 3 $/\$
5		elex 2566	. . . 4
6		elex 2566	. . . 4
7	5, 6	anim12i 321	. . 3 $/\$ $/\$
8	4, 7	syl 14	. 2 $/\$
9		nqex 6461	. . . . 5
10	9	ssex 3894	. . . 4
11	9	ssex 3894	. . . 4
12	10, 11	anim12i 321	. . 3 $/\$ $/\$
13	12	ad2antrr 457	. 2 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$ $/\$
14		df-inp 6564	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$
15	14	eleq2i 2104	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$
16		sseq1 2966	. . . . . . 7
17	16	anbi1d 438	. . . . . 6 $/\$ $/\$
18		eleq2 2101	. . . . . . . 8
19	18	rexbidv 2327	. . . . . . 7
20	19	anbi1d 438	. . . . . 6 $/\$ $/\$
21	17, 20	anbi12d 442	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
22		eleq2 2101	. . . . . . . . . . 11
23	22	anbi2d 437	. . . . . . . . . 10 $/\$ $/\$
24	23	rexbidv 2327	. . . . . . . . 9 $/\$ $/\$
25	18, 24	bibi12d 224	. . . . . . . 8 $/\$ $/\$
26	25	ralbidv 2326	. . . . . . 7 $/\$ $/\$
27	26	anbi1d 438	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
28	18	anbi1d 438	. . . . . . . 8 $/\$ $/\$
29	28	notbid 592	. . . . . . 7 $/\$ $/\$
30	29	ralbidv 2326	. . . . . 6 $/\$ $/\$
31	18	orbi1d 705	. . . . . . . 8 $\/$ $\/$
32	31	imbi2d 219	. . . . . . 7 $\/$ $\/$
33	32	2ralbidv 2348	. . . . . 6 $\/$ $\/$
34	27, 30, 33	3anbi123d 1207	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$
35	21, 34	anbi12d 442	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$
36		sseq1 2966	. . . . . . 7
37	36	anbi2d 437	. . . . . 6 $/\$ $/\$
38		eleq2 2101	. . . . . . . 8
39	38	rexbidv 2327	. . . . . . 7
40	39	anbi2d 437	. . . . . 6 $/\$ $/\$
41	37, 40	anbi12d 442	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
42		eleq2 2101	. . . . . . . . . . 11
43	42	anbi2d 437	. . . . . . . . . 10 $/\$ $/\$
44	43	rexbidv 2327	. . . . . . . . 9 $/\$ $/\$
45	38, 44	bibi12d 224	. . . . . . . 8 $/\$ $/\$
46	45	ralbidv 2326	. . . . . . 7 $/\$ $/\$
47	46	anbi2d 437	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
48	42	anbi2d 437	. . . . . . . 8 $/\$ $/\$
49	48	notbid 592	. . . . . . 7 $/\$ $/\$
50	49	ralbidv 2326	. . . . . 6 $/\$ $/\$
51	38	orbi2d 704	. . . . . . . 8 $\/$ $\/$
52	51	imbi2d 219	. . . . . . 7 $\/$ $\/$
53	52	2ralbidv 2348	. . . . . 6 $\/$ $\/$
54	47, 50, 53	3anbi123d 1207	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$
55	41, 54	anbi12d 442	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$
56	35, 55	opelopabg 4005	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$
57	15, 56	syl5bb 181	. 2 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$
58	8, 13, 57	pm5.21nii 620	1 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$

Colors of variables: wff set class

Syntax hints:

wn 3

wi 4 $/\$ wa 97

wb 98 $\/$ wo 629 $/\$ w3a 885

wceq 1243

wcel 1393

wral 2306

wrex 2307

cvv 2557

wss 2917

cpw 3359

cop 3378 class class class wbr 3764

copab 3817

cxp 4343

cnq 6378

cltq 6383

cnp 6389

This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-13 1404 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-coll 3872 ax-sep 3875 ax-pow 3927 ax-pr 3944 ax-un 4170 ax-iinf 4311

This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ral 2311 df-rex 2312 df-reu 2313 df-rab 2315 df-v 2559 df-sbc 2765 df-csb 2853 df-dif 2920 df-un 2922 df-in 2924 df-ss 2931 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-uni 3581 df-int 3616 df-iun 3659 df-br 3765 df-opab 3819 df-mpt 3820 df-id 4030 df-iom 4314 df-xp 4351 df-rel 4352 df-cnv 4353 df-co 4354 df-dm 4355 df-rn 4356 df-res 4357 df-ima 4358 df-iota 4867 df-fun 4904 df-fn 4905 df-f 4906 df-f1 4907 df-fo 4908 df-f1o 4909 df-fv 4910 df-qs 6112 df-ni 6402 df-nqqs 6446 df-inp 6564

This theorem is referenced by: elnp1st2nd 6574 prml 6575 prmu 6576 prssnql 6577 prssnqu 6578 prcdnql 6582 prcunqu 6583 prltlu 6585 prnmaxl 6586 prnminu 6587 prloc 6589 prdisj 6590 nqprxx 6644

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