elinp - Intuitionistic Logic Explorer

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Theorem elinp 7587

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 7584	. . . . 5
2	1	sseli 3189	. . . 4
3		opelxp 4705	. . . 4 $/\$
4	2, 3	sylib 122	. . 3 $/\$
5		elex 2783	. . . 4
6		elex 2783	. . . 4
7	5, 6	anim12i 338	. . 3 $/\$ $/\$
8	4, 7	syl 14	. 2 $/\$
9		nqex 7476	. . . . 5
10	9	ssex 4181	. . . 4
11	9	ssex 4181	. . . 4
12	10, 11	anim12i 338	. . 3 $/\$ $/\$
13	12	ad2antrr 488	. 2 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$ $/\$
14		df-inp 7579	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$
15	14	eleq2i 2272	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$
16		sseq1 3216	. . . . . . 7
17	16	anbi1d 465	. . . . . 6 $/\$ $/\$
18		eleq2 2269	. . . . . . . 8
19	18	rexbidv 2507	. . . . . . 7
20	19	anbi1d 465	. . . . . 6 $/\$ $/\$
21	17, 20	anbi12d 473	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
22		eleq2 2269	. . . . . . . . . . 11
23	22	anbi2d 464	. . . . . . . . . 10 $/\$ $/\$
24	23	rexbidv 2507	. . . . . . . . 9 $/\$ $/\$
25	18, 24	bibi12d 235	. . . . . . . 8 $/\$ $/\$
26	25	ralbidv 2506	. . . . . . 7 $/\$ $/\$
27	26	anbi1d 465	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
28	18	anbi1d 465	. . . . . . . 8 $/\$ $/\$
29	28	notbid 669	. . . . . . 7 $/\$ $/\$
30	29	ralbidv 2506	. . . . . 6 $/\$ $/\$
31	18	orbi1d 793	. . . . . . . 8 $\/$ $\/$
32	31	imbi2d 230	. . . . . . 7 $\/$ $\/$
33	32	2ralbidv 2530	. . . . . 6 $\/$ $\/$
34	27, 30, 33	3anbi123d 1325	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$
35	21, 34	anbi12d 473	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$
36		sseq1 3216	. . . . . . 7
37	36	anbi2d 464	. . . . . 6 $/\$ $/\$
38		eleq2 2269	. . . . . . . 8
39	38	rexbidv 2507	. . . . . . 7
40	39	anbi2d 464	. . . . . 6 $/\$ $/\$
41	37, 40	anbi12d 473	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
42		eleq2 2269	. . . . . . . . . . 11
43	42	anbi2d 464	. . . . . . . . . 10 $/\$ $/\$
44	43	rexbidv 2507	. . . . . . . . 9 $/\$ $/\$
45	38, 44	bibi12d 235	. . . . . . . 8 $/\$ $/\$
46	45	ralbidv 2506	. . . . . . 7 $/\$ $/\$
47	46	anbi2d 464	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
48	42	anbi2d 464	. . . . . . . 8 $/\$ $/\$
49	48	notbid 669	. . . . . . 7 $/\$ $/\$
50	49	ralbidv 2506	. . . . . 6 $/\$ $/\$
51	38	orbi2d 792	. . . . . . . 8 $\/$ $\/$
52	51	imbi2d 230	. . . . . . 7 $\/$ $\/$
53	52	2ralbidv 2530	. . . . . 6 $\/$ $\/$
54	47, 50, 53	3anbi123d 1325	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$
55	41, 54	anbi12d 473	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$
56	35, 55	opelopabg 4314	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$
57	15, 56	bitrid 192	. 2 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$
58	8, 13, 57	pm5.21nii 706	1 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$

Colors of variables: wff set class

Syntax hints:

wn 3

wi 4 $/\$ wa 104

wb 105 $\/$ wo 710 $/\$ w3a 981

wceq 1373

wcel 2176

wral 2484

wrex 2485

cvv 2772

wss 3166

cpw 3616

cop 3636 class class class wbr 4044

copab 4104

cxp 4673

cnq 7393

cltq 7398

cnp 7404

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 711 ax-5 1470 ax-7 1471 ax-gen 1472 ax-ie1 1516 ax-ie2 1517 ax-8 1527 ax-10 1528 ax-11 1529 ax-i12 1530 ax-bndl 1532 ax-4 1533 ax-17 1549 ax-i9 1553 ax-ial 1557 ax-i5r 1558 ax-13 2178 ax-14 2179 ax-ext 2187 ax-coll 4159 ax-sep 4162 ax-pow 4218 ax-pr 4253 ax-un 4480 ax-iinf 4636

This theorem depends on definitions: df-bi 117 df-3an 983 df-tru 1376 df-nf 1484 df-sb 1786 df-eu 2057 df-mo 2058 df-clab 2192 df-cleq 2198 df-clel 2201 df-nfc 2337 df-ral 2489 df-rex 2490 df-reu 2491 df-rab 2493 df-v 2774 df-sbc 2999 df-csb 3094 df-dif 3168 df-un 3170 df-in 3172 df-ss 3179 df-pw 3618 df-sn 3639 df-pr 3640 df-op 3642 df-uni 3851 df-int 3886 df-iun 3929 df-br 4045 df-opab 4106 df-mpt 4107 df-id 4340 df-iom 4639 df-xp 4681 df-rel 4682 df-cnv 4683 df-co 4684 df-dm 4685 df-rn 4686 df-res 4687 df-ima 4688 df-iota 5232 df-fun 5273 df-fn 5274 df-f 5275 df-f1 5276 df-fo 5277 df-f1o 5278 df-fv 5279 df-qs 6626 df-ni 7417 df-nqqs 7461 df-inp 7579

This theorem is referenced by: elnp1st2nd 7589 prml 7590 prmu 7591 prssnql 7592 prssnqu 7593 prcdnql 7597 prcunqu 7598 prltlu 7600 prnmaxl 7601 prnminu 7602 prloc 7604 prdisj 7605 nqprxx 7659 suplocexprlemex 7835

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