elnp1st2nd - Intuitionistic Logic Explorer

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Theorem elnp1st2nd 7536

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 7531	. . . . 5
2	1	sseli 3175	. . . 4
3		prop 7535	. . . . . . 7
4		elinp 7534	. . . . . . 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 6228	. . . 4
12	11	ad2antrr 488	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$
13		xp1st 6218	. . . . . . . 8
14	13	elpwid 3612	. . . . . . 7
15		xp2nd 6219	. . . . . . . 8
16	15	elpwid 3612	. . . . . . 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 2270	. 2 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$
22	10, 21	impbii 126	1 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$

Colors of variables: wff set class

Syntax hints:

wn 3

wi 4 $/\$ wa 104

wb 105 $\/$ wo 709 $/\$ w3a 980

wceq 1364

wcel 2164

wral 2472

wrex 2473

wss 3153

cpw 3601

cop 3621 class class class wbr 4029

cxp 4657

cfv 5254

c1st 6191

c2nd 6192

cnq 7340

cltq 7345

cnp 7351

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 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2166 ax-14 2167 ax-ext 2175 ax-coll 4144 ax-sep 4147 ax-pow 4203 ax-pr 4238 ax-un 4464 ax-iinf 4620

This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-nf 1472 df-sb 1774 df-eu 2045 df-mo 2046 df-clab 2180 df-cleq 2186 df-clel 2189 df-nfc 2325 df-ral 2477 df-rex 2478 df-reu 2479 df-rab 2481 df-v 2762 df-sbc 2986 df-csb 3081 df-dif 3155 df-un 3157 df-in 3159 df-ss 3166 df-pw 3603 df-sn 3624 df-pr 3625 df-op 3627 df-uni 3836 df-int 3871 df-iun 3914 df-br 4030 df-opab 4091 df-mpt 4092 df-id 4324 df-iom 4623 df-xp 4665 df-rel 4666 df-cnv 4667 df-co 4668 df-dm 4669 df-rn 4670 df-res 4671 df-ima 4672 df-iota 5215 df-fun 5256 df-fn 5257 df-f 5258 df-f1 5259 df-fo 5260 df-f1o 5261 df-fv 5262 df-1st 6193 df-2nd 6194 df-qs 6593 df-ni 7364 df-nqqs 7408 df-inp 7526

This theorem is referenced by: addclpr 7597 mulclpr 7632 ltexprlempr 7668 recexprlempr 7692 cauappcvgprlemcl 7713 caucvgprlemcl 7736 caucvgprprlemcl 7764