elinp - Intuitionistic Logic Explorer

	Intuitionistic Logic Explorer		< Previous Next > Nearby theorems
Mirrors > Home > ILE Home > Th. List > elinp		Unicode version

Theorem elinp 7429

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 7426	. . . . 5
2	1	sseli 3143	. . . 4
3		opelxp 4639	. . . 4 $/\$
4	2, 3	sylib 121	. . 3 $/\$
5		elex 2741	. . . 4
6		elex 2741	. . . 4
7	5, 6	anim12i 336	. . 3 $/\$ $/\$
8	4, 7	syl 14	. 2 $/\$
9		nqex 7318	. . . . 5
10	9	ssex 4124	. . . 4
11	9	ssex 4124	. . . 4
12	10, 11	anim12i 336	. . 3 $/\$ $/\$
13	12	ad2antrr 485	. 2 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$ $/\$
14		df-inp 7421	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$
15	14	eleq2i 2237	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$
16		sseq1 3170	. . . . . . 7
17	16	anbi1d 462	. . . . . 6 $/\$ $/\$
18		eleq2 2234	. . . . . . . 8
19	18	rexbidv 2471	. . . . . . 7
20	19	anbi1d 462	. . . . . 6 $/\$ $/\$
21	17, 20	anbi12d 470	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
22		eleq2 2234	. . . . . . . . . . 11
23	22	anbi2d 461	. . . . . . . . . 10 $/\$ $/\$
24	23	rexbidv 2471	. . . . . . . . 9 $/\$ $/\$
25	18, 24	bibi12d 234	. . . . . . . 8 $/\$ $/\$
26	25	ralbidv 2470	. . . . . . 7 $/\$ $/\$
27	26	anbi1d 462	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
28	18	anbi1d 462	. . . . . . . 8 $/\$ $/\$
29	28	notbid 662	. . . . . . 7 $/\$ $/\$
30	29	ralbidv 2470	. . . . . 6 $/\$ $/\$
31	18	orbi1d 786	. . . . . . . 8 $\/$ $\/$
32	31	imbi2d 229	. . . . . . 7 $\/$ $\/$
33	32	2ralbidv 2494	. . . . . 6 $\/$ $\/$
34	27, 30, 33	3anbi123d 1307	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$
35	21, 34	anbi12d 470	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$
36		sseq1 3170	. . . . . . 7
37	36	anbi2d 461	. . . . . 6 $/\$ $/\$
38		eleq2 2234	. . . . . . . 8
39	38	rexbidv 2471	. . . . . . 7
40	39	anbi2d 461	. . . . . 6 $/\$ $/\$
41	37, 40	anbi12d 470	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
42		eleq2 2234	. . . . . . . . . . 11
43	42	anbi2d 461	. . . . . . . . . 10 $/\$ $/\$
44	43	rexbidv 2471	. . . . . . . . 9 $/\$ $/\$
45	38, 44	bibi12d 234	. . . . . . . 8 $/\$ $/\$
46	45	ralbidv 2470	. . . . . . 7 $/\$ $/\$
47	46	anbi2d 461	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
48	42	anbi2d 461	. . . . . . . 8 $/\$ $/\$
49	48	notbid 662	. . . . . . 7 $/\$ $/\$
50	49	ralbidv 2470	. . . . . 6 $/\$ $/\$
51	38	orbi2d 785	. . . . . . . 8 $\/$ $\/$
52	51	imbi2d 229	. . . . . . 7 $\/$ $\/$
53	52	2ralbidv 2494	. . . . . 6 $\/$ $\/$
54	47, 50, 53	3anbi123d 1307	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$
55	41, 54	anbi12d 470	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$
56	35, 55	opelopabg 4251	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$
57	15, 56	syl5bb 191	. 2 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$
58	8, 13, 57	pm5.21nii 699	1 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$

Colors of variables: wff set class

Syntax hints:

wn 3

wi 4 $/\$ wa 103

wb 104 $\/$ wo 703 $/\$ w3a 973

wceq 1348

wcel 2141

wral 2448

wrex 2449

cvv 2730

wss 3121

cpw 3564

cop 3584 class class class wbr 3987

copab 4047

cxp 4607

cnq 7235

cltq 7240

cnp 7246

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-coll 4102 ax-sep 4105 ax-pow 4158 ax-pr 4192 ax-un 4416 ax-iinf 4570

This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ral 2453 df-rex 2454 df-reu 2455 df-rab 2457 df-v 2732 df-sbc 2956 df-csb 3050 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-pw 3566 df-sn 3587 df-pr 3588 df-op 3590 df-uni 3795 df-int 3830 df-iun 3873 df-br 3988 df-opab 4049 df-mpt 4050 df-id 4276 df-iom 4573 df-xp 4615 df-rel 4616 df-cnv 4617 df-co 4618 df-dm 4619 df-rn 4620 df-res 4621 df-ima 4622 df-iota 5158 df-fun 5198 df-fn 5199 df-f 5200 df-f1 5201 df-fo 5202 df-f1o 5203 df-fv 5204 df-qs 6517 df-ni 7259 df-nqqs 7303 df-inp 7421

This theorem is referenced by: elnp1st2nd 7431 prml 7432 prmu 7433 prssnql 7434 prssnqu 7435 prcdnql 7439 prcunqu 7440 prltlu 7442 prnmaxl 7443 prnminu 7444 prloc 7446 prdisj 7447 nqprxx 7501 suplocexprlemex 7677

W3C validator