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Theorem elinp 7558

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 7555	. . . . 5
2	1	sseli 3180	. . . 4
3		opelxp 4694	. . . 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 7447	. . . . 5
10	9	ssex 4171	. . . 4
11	9	ssex 4171	. . . 4
12	10, 11	anim12i 338	. . 3 $/\$ $/\$
13	12	ad2antrr 488	. 2 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$ $/\$
14		df-inp 7550	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$
15	14	eleq2i 2263	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$
16		sseq1 3207	. . . . . . 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 3207	. . . . . . 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 4303	. . 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 3606

cop 3626 class class class wbr 4034

copab 4094

cxp 4662

cnq 7364

cltq 7369

cnp 7375

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 4149 ax-sep 4152 ax-pow 4208 ax-pr 4243 ax-un 4469 ax-iinf 4625

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 3608 df-sn 3629 df-pr 3630 df-op 3632 df-uni 3841 df-int 3876 df-iun 3919 df-br 4035 df-opab 4096 df-mpt 4097 df-id 4329 df-iom 4628 df-xp 4670 df-rel 4671 df-cnv 4672 df-co 4673 df-dm 4674 df-rn 4675 df-res 4676 df-ima 4677 df-iota 5220 df-fun 5261 df-fn 5262 df-f 5263 df-f1 5264 df-fo 5265 df-f1o 5266 df-fv 5267 df-qs 6607 df-ni 7388 df-nqqs 7432 df-inp 7550

This theorem is referenced by: elnp1st2nd 7560 prml 7561 prmu 7562 prssnql 7563 prssnqu 7564 prcdnql 7568 prcunqu 7569 prltlu 7571 prnmaxl 7572 prnminu 7573 prloc 7575 prdisj 7576 nqprxx 7630 suplocexprlemex 7806

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