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Theorem elinp 7841

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 7838	. . . . 5
2	1	sseli 3244	. . . 4
3		opelxp 4804	. . . 4 $/\$
4	2, 3	sylib 122	. . 3 $/\$
5		elex 2833	. . . 4
6		elex 2833	. . . 4
7	5, 6	anim12i 338	. . 3 $/\$ $/\$
8	4, 7	syl 14	. 2 $/\$
9		nqex 7730	. . . . 5
10	9	ssex 4270	. . . 4
11	9	ssex 4270	. . . 4
12	10, 11	anim12i 338	. . 3 $/\$ $/\$
13	12	ad2antrr 492	. 2 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$ $/\$
14		df-inp 7833	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$
15	14	eleq2i 2305	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$
16		sseq1 3271	. . . . . . 7
17	16	anbi1d 469	. . . . . 6 $/\$ $/\$
18		eleq2 2302	. . . . . . . 8
19	18	rexbidv 2551	. . . . . . 7
20	19	anbi1d 469	. . . . . 6 $/\$ $/\$
21	17, 20	anbi12d 477	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
22		eleq2 2302	. . . . . . . . . . 11
23	22	anbi2d 468	. . . . . . . . . 10 $/\$ $/\$
24	23	rexbidv 2551	. . . . . . . . 9 $/\$ $/\$
25	18, 24	bibi12d 235	. . . . . . . 8 $/\$ $/\$
26	25	ralbidv 2550	. . . . . . 7 $/\$ $/\$
27	26	anbi1d 469	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
28	18	anbi1d 469	. . . . . . . 8 $/\$ $/\$
29	28	notbid 677	. . . . . . 7 $/\$ $/\$
30	29	ralbidv 2550	. . . . . 6 $/\$ $/\$
31	18	orbi1d 803	. . . . . . . 8 $\/$ $\/$
32	31	imbi2d 230	. . . . . . 7 $\/$ $\/$
33	32	2ralbidv 2574	. . . . . 6 $\/$ $\/$
34	27, 30, 33	3anbi123d 1353	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$
35	21, 34	anbi12d 477	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$
36		sseq1 3271	. . . . . . 7
37	36	anbi2d 468	. . . . . 6 $/\$ $/\$
38		eleq2 2302	. . . . . . . 8
39	38	rexbidv 2551	. . . . . . 7
40	39	anbi2d 468	. . . . . 6 $/\$ $/\$
41	37, 40	anbi12d 477	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
42		eleq2 2302	. . . . . . . . . . 11
43	42	anbi2d 468	. . . . . . . . . 10 $/\$ $/\$
44	43	rexbidv 2551	. . . . . . . . 9 $/\$ $/\$
45	38, 44	bibi12d 235	. . . . . . . 8 $/\$ $/\$
46	45	ralbidv 2550	. . . . . . 7 $/\$ $/\$
47	46	anbi2d 468	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
48	42	anbi2d 468	. . . . . . . 8 $/\$ $/\$
49	48	notbid 677	. . . . . . 7 $/\$ $/\$
50	49	ralbidv 2550	. . . . . 6 $/\$ $/\$
51	38	orbi2d 802	. . . . . . . 8 $\/$ $\/$
52	51	imbi2d 230	. . . . . . 7 $\/$ $\/$
53	52	2ralbidv 2574	. . . . . 6 $\/$ $\/$
54	47, 50, 53	3anbi123d 1353	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$
55	41, 54	anbi12d 477	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$
56	35, 55	opelopabg 4410	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$
57	15, 56	bitrid 192	. 2 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$
58	8, 13, 57	pm5.21nii 716	1 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$

Colors of variables: wff set class

This proof depends on syntax axioms:

wn 3

wi 4 $/\$ wa 104

wb 105 $\/$ wo 720 $/\$ w3a 1009

wceq 1402

wcel 2209

wral 2528

wrex 2529

cvv 2821

wss 3220

cpw 3688

cop 3712 class class class wbr 4130

copab 4191

cxp 4772

cnq 7647

cltq 7652

cnp 7658

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 623 ax-in2 624 ax-io 721 ax-5 1500 ax-7 1501 ax-gen 1502 ax-ie1 1546 ax-ie2 1547 ax-8 1557 ax-10 1558 ax-11 1559 ax-i12 1560 ax-bndl 1562 ax-4 1563 ax-17 1579 ax-i9 1583 ax-ial 1587 ax-i5r 1588 ax-14 2212 ax-ext 2220 ax-coll 4246 ax-sep 4249 ax-pow 4311 ax-pr 4346 ax-un 4578 ax-iinf 4735

This proof depends on definitions: df-bi 117 df-3an 1011 df-tru 1405 df-nf 1514 df-sb 1816 df-eu 2089 df-mo 2090 df-clab 2225 df-cleq 2231 df-clel 2234 df-nfc 2381 df-ral 2533 df-rex 2534 df-reu 2535 df-rab 2537 df-v 2823 df-sbc 3052 df-csb 3148 df-dif 3222 df-un 3224 df-in 3226 df-ss 3233 df-pw 3690 df-sn 3715 df-pr 3716 df-op 3718 df-uni 3936 df-int 3971 df-iun 4014 df-br 4131 df-opab 4193 df-mpt 4194 df-id 4438 df-iom 4738 df-xp 4780 df-rel 4781 df-cnv 4782 df-co 4783 df-dm 4784 df-rn 4785 df-res 4786 df-ima 4787 df-iota 5337 df-fun 5379 df-fn 5380 df-f 5381 df-f1 5382 df-fo 5383 df-f1o 5384 df-fv 5385 df-qs 6813 df-ni 7671 df-nqqs 7715 df-inp 7833

This theorem is used by: elnp1st2nd 7843 prml 7844 prmu 7845 prssnql 7846 prssnqu 7847 prcdnql 7851 prcunqu 7852 prltlu 7854 prnmaxl 7855 prnminu 7856 prloc 7858 prdisj 7859 nqprxx 7913 suplocexprlemex 8089

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