elinp - Intuitionistic Logic Explorer

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Theorem elinp 7541

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 7538	. . . . 5
2	1	sseli 3179	. . . 4
3		opelxp 4693	. . . 4 $/\$
4	2, 3	sylib 122	. . 3 $/\$
5		elex 2774	. . . 4
6		elex 2774	. . . 4
7	5, 6	anim12i 338	. . 3 $/\$ $/\$
8	4, 7	syl 14	. 2 $/\$
9		nqex 7430	. . . . 5
10	9	ssex 4170	. . . 4
11	9	ssex 4170	. . . 4
12	10, 11	anim12i 338	. . 3 $/\$ $/\$
13	12	ad2antrr 488	. 2 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$ $/\$
14		df-inp 7533	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$
15	14	eleq2i 2263	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$
16		sseq1 3206	. . . . . . 7
17	16	anbi1d 465	. . . . . 6 $/\$ $/\$
18		eleq2 2260	. . . . . . . 8
19	18	rexbidv 2498	. . . . . . 7
20	19	anbi1d 465	. . . . . 6 $/\$ $/\$
21	17, 20	anbi12d 473	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
22		eleq2 2260	. . . . . . . . . . 11
23	22	anbi2d 464	. . . . . . . . . 10 $/\$ $/\$
24	23	rexbidv 2498	. . . . . . . . 9 $/\$ $/\$
25	18, 24	bibi12d 235	. . . . . . . 8 $/\$ $/\$
26	25	ralbidv 2497	. . . . . . 7 $/\$ $/\$
27	26	anbi1d 465	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
28	18	anbi1d 465	. . . . . . . 8 $/\$ $/\$
29	28	notbid 668	. . . . . . 7 $/\$ $/\$
30	29	ralbidv 2497	. . . . . 6 $/\$ $/\$
31	18	orbi1d 792	. . . . . . . 8 $\/$ $\/$
32	31	imbi2d 230	. . . . . . 7 $\/$ $\/$
33	32	2ralbidv 2521	. . . . . 6 $\/$ $\/$
34	27, 30, 33	3anbi123d 1323	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$
35	21, 34	anbi12d 473	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$
36		sseq1 3206	. . . . . . 7
37	36	anbi2d 464	. . . . . 6 $/\$ $/\$
38		eleq2 2260	. . . . . . . 8
39	38	rexbidv 2498	. . . . . . 7
40	39	anbi2d 464	. . . . . 6 $/\$ $/\$
41	37, 40	anbi12d 473	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
42		eleq2 2260	. . . . . . . . . . 11
43	42	anbi2d 464	. . . . . . . . . 10 $/\$ $/\$
44	43	rexbidv 2498	. . . . . . . . 9 $/\$ $/\$
45	38, 44	bibi12d 235	. . . . . . . 8 $/\$ $/\$
46	45	ralbidv 2497	. . . . . . 7 $/\$ $/\$
47	46	anbi2d 464	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
48	42	anbi2d 464	. . . . . . . 8 $/\$ $/\$
49	48	notbid 668	. . . . . . 7 $/\$ $/\$
50	49	ralbidv 2497	. . . . . 6 $/\$ $/\$
51	38	orbi2d 791	. . . . . . . 8 $\/$ $\/$
52	51	imbi2d 230	. . . . . . 7 $\/$ $\/$
53	52	2ralbidv 2521	. . . . . 6 $\/$ $\/$
54	47, 50, 53	3anbi123d 1323	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$
55	41, 54	anbi12d 473	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$
56	35, 55	opelopabg 4302	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$
57	15, 56	bitrid 192	. 2 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$
58	8, 13, 57	pm5.21nii 705	1 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$

Colors of variables: wff set class

Syntax hints:

wn 3

wi 4 $/\$ wa 104

wb 105 $\/$ wo 709 $/\$ w3a 980

wceq 1364

wcel 2167

wral 2475

wrex 2476

cvv 2763

wss 3157

cpw 3605

cop 3625 class class class wbr 4033

copab 4093

cxp 4661

cnq 7347

cltq 7352

cnp 7358

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 615 ax-in2 616 ax-io 710 ax-5 1461 ax-7 1462 ax-gen 1463 ax-ie1 1507 ax-ie2 1508 ax-8 1518 ax-10 1519 ax-11 1520 ax-i12 1521 ax-bndl 1523 ax-4 1524 ax-17 1540 ax-i9 1544 ax-ial 1548 ax-i5r 1549 ax-13 2169 ax-14 2170 ax-ext 2178 ax-coll 4148 ax-sep 4151 ax-pow 4207 ax-pr 4242 ax-un 4468 ax-iinf 4624

This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-nf 1475 df-sb 1777 df-eu 2048 df-mo 2049 df-clab 2183 df-cleq 2189 df-clel 2192 df-nfc 2328 df-ral 2480 df-rex 2481 df-reu 2482 df-rab 2484 df-v 2765 df-sbc 2990 df-csb 3085 df-dif 3159 df-un 3161 df-in 3163 df-ss 3170 df-pw 3607 df-sn 3628 df-pr 3629 df-op 3631 df-uni 3840 df-int 3875 df-iun 3918 df-br 4034 df-opab 4095 df-mpt 4096 df-id 4328 df-iom 4627 df-xp 4669 df-rel 4670 df-cnv 4671 df-co 4672 df-dm 4673 df-rn 4674 df-res 4675 df-ima 4676 df-iota 5219 df-fun 5260 df-fn 5261 df-f 5262 df-f1 5263 df-fo 5264 df-f1o 5265 df-fv 5266 df-qs 6598 df-ni 7371 df-nqqs 7415 df-inp 7533

This theorem is referenced by: elnp1st2nd 7543 prml 7544 prmu 7545 prssnql 7546 prssnqu 7547 prcdnql 7551 prcunqu 7552 prltlu 7554 prnmaxl 7555 prnminu 7556 prloc 7558 prdisj 7559 nqprxx 7613 suplocexprlemex 7789

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