elinp - Intuitionistic Logic Explorer

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Theorem elinp 7436

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 7433	. . . . 5
2	1	sseli 3143	. . . 4
3		opelxp 4641	. . . 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 7325	. . . . 5
10	9	ssex 4126	. . . 4
11	9	ssex 4126	. . . 4
12	10, 11	anim12i 336	. . 3 $/\$ $/\$
13	12	ad2antrr 485	. 2 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$ $/\$
14		df-inp 7428	. . . 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 4253	. . 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 3566

cop 3586 class class class wbr 3989

copab 4049

cxp 4609

cnq 7242

cltq 7247

cnp 7253

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 4104 ax-sep 4107 ax-pow 4160 ax-pr 4194 ax-un 4418 ax-iinf 4572

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 3568 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-int 3832 df-iun 3875 df-br 3990 df-opab 4051 df-mpt 4052 df-id 4278 df-iom 4575 df-xp 4617 df-rel 4618 df-cnv 4619 df-co 4620 df-dm 4621 df-rn 4622 df-res 4623 df-ima 4624 df-iota 5160 df-fun 5200 df-fn 5201 df-f 5202 df-f1 5203 df-fo 5204 df-f1o 5205 df-fv 5206 df-qs 6519 df-ni 7266 df-nqqs 7310 df-inp 7428

This theorem is referenced by: elnp1st2nd 7438 prml 7439 prmu 7440 prssnql 7441 prssnqu 7442 prcdnql 7446 prcunqu 7447 prltlu 7449 prnmaxl 7450 prnminu 7451 prloc 7453 prdisj 7454 nqprxx 7508 suplocexprlemex 7684

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