elnp1st2nd - Intuitionistic Logic Explorer

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Theorem elnp1st2nd 7396

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 7391	. . . . 5
2	1	sseli 3124	. . . 4
3		prop 7395	. . . . . . 7
4		elinp 7394	. . . . . . 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 6123	. . . 4
12	11	ad2antrr 480	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$
13		xp1st 6113	. . . . . . . 8
14	13	elpwid 3554	. . . . . . 7
15		xp2nd 6114	. . . . . . . 8
16	15	elpwid 3554	. . . . . . 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 2234	. 2 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$
22	10, 21	impbii 125	1 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$

Colors of variables: wff set class

Syntax hints:

wn 3

wi 4 $/\$ wa 103

wb 104 $\/$ wo 698 $/\$ w3a 963

wceq 1335

wcel 2128

wral 2435

wrex 2436

wss 3102

cpw 3543

cop 3563 class class class wbr 3965

cxp 4584

cfv 5170

c1st 6086

c2nd 6087

cnq 7200

cltq 7205

cnp 7211

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 604 ax-in2 605 ax-io 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-13 2130 ax-14 2131 ax-ext 2139 ax-coll 4079 ax-sep 4082 ax-pow 4135 ax-pr 4169 ax-un 4393 ax-iinf 4547

This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1338 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ral 2440 df-rex 2441 df-reu 2442 df-rab 2444 df-v 2714 df-sbc 2938 df-csb 3032 df-dif 3104 df-un 3106 df-in 3108 df-ss 3115 df-pw 3545 df-sn 3566 df-pr 3567 df-op 3569 df-uni 3773 df-int 3808 df-iun 3851 df-br 3966 df-opab 4026 df-mpt 4027 df-id 4253 df-iom 4550 df-xp 4592 df-rel 4593 df-cnv 4594 df-co 4595 df-dm 4596 df-rn 4597 df-res 4598 df-ima 4599 df-iota 5135 df-fun 5172 df-fn 5173 df-f 5174 df-f1 5175 df-fo 5176 df-f1o 5177 df-fv 5178 df-1st 6088 df-2nd 6089 df-qs 6486 df-ni 7224 df-nqqs 7268 df-inp 7386

This theorem is referenced by: addclpr 7457 mulclpr 7492 ltexprlempr 7528 recexprlempr 7552 cauappcvgprlemcl 7573 caucvgprlemcl 7596 caucvgprprlemcl 7624