elnp1st2nd - Intuitionistic Logic Explorer

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Theorem elnp1st2nd 7756

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 7751	. . . . 5
2	1	sseli 3224	. . . 4
3		prop 7755	. . . . . . 7
4		elinp 7754	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$
5	3, 4	sylib 122	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$
6	5	simpld 112	. . . . 5 $/\$ $/\$ $/\$
7	6	simprd 114	. . . 4 $/\$
8	2, 7	jca 306	. . 3 $/\$ $/\$
9	5	simprd 114	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$
10	8, 9	jca 306	. 2 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$
11		1st2nd2 6347	. . . 4
12	11	ad2antrr 488	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$
13		xp1st 6337	. . . . . . . 8
14	13	elpwid 3667	. . . . . . 7
15		xp2nd 6338	. . . . . . . 8
16	15	elpwid 3667	. . . . . . 7
17	14, 16	jca 306	. . . . . 6 $/\$
18	17	anim1i 340	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
19	18	anim1i 340	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$
20	19, 4	sylibr 134	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$
21	12, 20	eqeltrd 2308	. 2 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$
22	10, 21	impbii 126	1 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$

Colors of variables: wff set class

Syntax hints:

wn 3

wi 4 $/\$ wa 104

wb 105 $\/$ wo 716 $/\$ w3a 1005

wceq 1398

wcel 2202

wral 2511

wrex 2512

wss 3201

cpw 3656

cop 3676 class class class wbr 4093

cxp 4729

cfv 5333

c1st 6310

c2nd 6311

cnq 7560

cltq 7565

cnp 7571

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 619 ax-in2 620 ax-io 717 ax-5 1496 ax-7 1497 ax-gen 1498 ax-ie1 1542 ax-ie2 1543 ax-8 1553 ax-10 1554 ax-11 1555 ax-i12 1556 ax-bndl 1558 ax-4 1559 ax-17 1575 ax-i9 1579 ax-ial 1583 ax-i5r 1584 ax-13 2204 ax-14 2205 ax-ext 2213 ax-coll 4209 ax-sep 4212 ax-pow 4270 ax-pr 4305 ax-un 4536 ax-iinf 4692

This theorem depends on definitions: df-bi 117 df-3an 1007 df-tru 1401 df-nf 1510 df-sb 1811 df-eu 2082 df-mo 2083 df-clab 2218 df-cleq 2224 df-clel 2227 df-nfc 2364 df-ral 2516 df-rex 2517 df-reu 2518 df-rab 2520 df-v 2805 df-sbc 3033 df-csb 3129 df-dif 3203 df-un 3205 df-in 3207 df-ss 3214 df-pw 3658 df-sn 3679 df-pr 3680 df-op 3682 df-uni 3899 df-int 3934 df-iun 3977 df-br 4094 df-opab 4156 df-mpt 4157 df-id 4396 df-iom 4695 df-xp 4737 df-rel 4738 df-cnv 4739 df-co 4740 df-dm 4741 df-rn 4742 df-res 4743 df-ima 4744 df-iota 5293 df-fun 5335 df-fn 5336 df-f 5337 df-f1 5338 df-fo 5339 df-f1o 5340 df-fv 5341 df-1st 6312 df-2nd 6313 df-qs 6751 df-ni 7584 df-nqqs 7628 df-inp 7746

This theorem is referenced by: addclpr 7817 mulclpr 7852 ltexprlempr 7888 recexprlempr 7912 cauappcvgprlemcl 7933 caucvgprlemcl 7956 caucvgprprlemcl 7984