elinp - Intuitionistic Logic Explorer

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Theorem elinp 7831

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 7828	. . . . 5
2	1	sseli 3244	. . . 4
3		opelxp 4799	. . . 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 7720	. . . . 5
10	9	ssex 4265	. . . 4
11	9	ssex 4265	. . . 4
12	10, 11	anim12i 338	. . 3 $/\$ $/\$
13	12	ad2antrr 492	. 2 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$ $/\$
14		df-inp 7823	. . . 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 4405	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$
57	15, 56	bitrid 192	. 2 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$
58	8, 13, 57	pm5.21nii 716	1 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$

Colors of variables: wff set class

Syntax hints:

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 3685

cop 3708 class class class wbr 4125

copab 4186

cxp 4767

cnq 7637

cltq 7642

cnp 7648

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 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 4241 ax-sep 4244 ax-pow 4306 ax-pr 4341 ax-un 4573 ax-iinf 4730

This theorem 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 3687 df-sn 3711 df-pr 3712 df-op 3714 df-uni 3931 df-int 3966 df-iun 4009 df-br 4126 df-opab 4188 df-mpt 4189 df-id 4433 df-iom 4733 df-xp 4775 df-rel 4776 df-cnv 4777 df-co 4778 df-dm 4779 df-rn 4780 df-res 4781 df-ima 4782 df-iota 5332 df-fun 5374 df-fn 5375 df-f 5376 df-f1 5377 df-fo 5378 df-f1o 5379 df-fv 5380 df-qs 6803 df-ni 7661 df-nqqs 7705 df-inp 7823

This theorem is referenced by: elnp1st2nd 7833 prml 7834 prmu 7835 prssnql 7836 prssnqu 7837 prcdnql 7841 prcunqu 7842 prltlu 7844 prnmaxl 7845 prnminu 7846 prloc 7848 prdisj 7849 nqprxx 7903 suplocexprlemex 8079

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