elnp1st2nd - Intuitionistic Logic Explorer

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Theorem elnp1st2nd 7284

Description: Membership in positive reals, using

and

to refer to the lower and upper cut. (Contributed by Jim Kingdon, 3-Oct-2019.)

**Assertion**
Ref	Expression
elnp1st2nd	$/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$

**Proof of Theorem elnp1st2nd**
Step	Hyp	Ref	Expression
1		npsspw 7279	. . . . 5
2	1	sseli 3093	. . . 4
3		prop 7283	. . . . . . 7
4		elinp 7282	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$
5	3, 4	sylib 121	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$
6	5	simpld 111	. . . . 5 $/\$ $/\$ $/\$
7	6	simprd 113	. . . 4 $/\$
8	2, 7	jca 304	. . 3 $/\$ $/\$
9	5	simprd 113	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$
10	8, 9	jca 304	. 2 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$
11		1st2nd2 6073	. . . 4
12	11	ad2antrr 479	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$
13		xp1st 6063	. . . . . . . 8
14	13	elpwid 3521	. . . . . . 7
15		xp2nd 6064	. . . . . . . 8
16	15	elpwid 3521	. . . . . . 7
17	14, 16	jca 304	. . . . . 6 $/\$
18	17	anim1i 338	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
19	18	anim1i 338	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$
20	19, 4	sylibr 133	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$
21	12, 20	eqeltrd 2216	. 2 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$
22	10, 21	impbii 125	1 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$

Colors of variables: wff set class

Syntax hints:

wn 3

wi 4 $/\$ wa 103

wb 104 $\/$ wo 697 $/\$ w3a 962

wceq 1331

wcel 1480

wral 2416

wrex 2417

wss 3071

cpw 3510

cop 3530 class class class wbr 3929

cxp 4537

cfv 5123

c1st 6036

c2nd 6037

cnq 7088

cltq 7093

cnp 7099

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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-coll 4043 ax-sep 4046 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-iinf 4502

This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-rex 2422 df-reu 2423 df-rab 2425 df-v 2688 df-sbc 2910 df-csb 3004 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-int 3772 df-iun 3815 df-br 3930 df-opab 3990 df-mpt 3991 df-id 4215 df-iom 4505 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-f1 5128 df-fo 5129 df-f1o 5130 df-fv 5131 df-1st 6038 df-2nd 6039 df-qs 6435 df-ni 7112 df-nqqs 7156 df-inp 7274

This theorem is referenced by: addclpr 7345 mulclpr 7380 ltexprlempr 7416 recexprlempr 7440 cauappcvgprlemcl 7461 caucvgprlemcl 7484 caucvgprprlemcl 7512