elinp - Intuitionistic Logic Explorer

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Theorem elinp 7475

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 7472	. . . . 5
2	1	sseli 3153	. . . 4
3		opelxp 4658	. . . 4 $/\$
4	2, 3	sylib 122	. . 3 $/\$
5		elex 2750	. . . 4
6		elex 2750	. . . 4
7	5, 6	anim12i 338	. . 3 $/\$ $/\$
8	4, 7	syl 14	. 2 $/\$
9		nqex 7364	. . . . 5
10	9	ssex 4142	. . . 4
11	9	ssex 4142	. . . 4
12	10, 11	anim12i 338	. . 3 $/\$ $/\$
13	12	ad2antrr 488	. 2 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$ $/\$
14		df-inp 7467	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$
15	14	eleq2i 2244	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$
16		sseq1 3180	. . . . . . 7
17	16	anbi1d 465	. . . . . 6 $/\$ $/\$
18		eleq2 2241	. . . . . . . 8
19	18	rexbidv 2478	. . . . . . 7
20	19	anbi1d 465	. . . . . 6 $/\$ $/\$
21	17, 20	anbi12d 473	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
22		eleq2 2241	. . . . . . . . . . 11
23	22	anbi2d 464	. . . . . . . . . 10 $/\$ $/\$
24	23	rexbidv 2478	. . . . . . . . 9 $/\$ $/\$
25	18, 24	bibi12d 235	. . . . . . . 8 $/\$ $/\$
26	25	ralbidv 2477	. . . . . . 7 $/\$ $/\$
27	26	anbi1d 465	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
28	18	anbi1d 465	. . . . . . . 8 $/\$ $/\$
29	28	notbid 667	. . . . . . 7 $/\$ $/\$
30	29	ralbidv 2477	. . . . . 6 $/\$ $/\$
31	18	orbi1d 791	. . . . . . . 8 $\/$ $\/$
32	31	imbi2d 230	. . . . . . 7 $\/$ $\/$
33	32	2ralbidv 2501	. . . . . 6 $\/$ $\/$
34	27, 30, 33	3anbi123d 1312	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$
35	21, 34	anbi12d 473	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$
36		sseq1 3180	. . . . . . 7
37	36	anbi2d 464	. . . . . 6 $/\$ $/\$
38		eleq2 2241	. . . . . . . 8
39	38	rexbidv 2478	. . . . . . 7
40	39	anbi2d 464	. . . . . 6 $/\$ $/\$
41	37, 40	anbi12d 473	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
42		eleq2 2241	. . . . . . . . . . 11
43	42	anbi2d 464	. . . . . . . . . 10 $/\$ $/\$
44	43	rexbidv 2478	. . . . . . . . 9 $/\$ $/\$
45	38, 44	bibi12d 235	. . . . . . . 8 $/\$ $/\$
46	45	ralbidv 2477	. . . . . . 7 $/\$ $/\$
47	46	anbi2d 464	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
48	42	anbi2d 464	. . . . . . . 8 $/\$ $/\$
49	48	notbid 667	. . . . . . 7 $/\$ $/\$
50	49	ralbidv 2477	. . . . . 6 $/\$ $/\$
51	38	orbi2d 790	. . . . . . . 8 $\/$ $\/$
52	51	imbi2d 230	. . . . . . 7 $\/$ $\/$
53	52	2ralbidv 2501	. . . . . 6 $\/$ $\/$
54	47, 50, 53	3anbi123d 1312	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$
55	41, 54	anbi12d 473	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$
56	35, 55	opelopabg 4270	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$
57	15, 56	bitrid 192	. 2 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$
58	8, 13, 57	pm5.21nii 704	1 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$

Colors of variables: wff set class

Syntax hints:

wn 3

wi 4 $/\$ wa 104

wb 105 $\/$ wo 708 $/\$ w3a 978

wceq 1353

wcel 2148

wral 2455

wrex 2456

cvv 2739

wss 3131

cpw 3577

cop 3597 class class class wbr 4005

copab 4065

cxp 4626

cnq 7281

cltq 7286

cnp 7292

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 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-coll 4120 ax-sep 4123 ax-pow 4176 ax-pr 4211 ax-un 4435 ax-iinf 4589

This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ral 2460 df-rex 2461 df-reu 2462 df-rab 2464 df-v 2741 df-sbc 2965 df-csb 3060 df-dif 3133 df-un 3135 df-in 3137 df-ss 3144 df-pw 3579 df-sn 3600 df-pr 3601 df-op 3603 df-uni 3812 df-int 3847 df-iun 3890 df-br 4006 df-opab 4067 df-mpt 4068 df-id 4295 df-iom 4592 df-xp 4634 df-rel 4635 df-cnv 4636 df-co 4637 df-dm 4638 df-rn 4639 df-res 4640 df-ima 4641 df-iota 5180 df-fun 5220 df-fn 5221 df-f 5222 df-f1 5223 df-fo 5224 df-f1o 5225 df-fv 5226 df-qs 6543 df-ni 7305 df-nqqs 7349 df-inp 7467

This theorem is referenced by: elnp1st2nd 7477 prml 7478 prmu 7479 prssnql 7480 prssnqu 7481 prcdnql 7485 prcunqu 7486 prltlu 7488 prnmaxl 7489 prnminu 7490 prloc 7492 prdisj 7493 nqprxx 7547 suplocexprlemex 7723

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