elnp1st2nd - Intuitionistic Logic Explorer

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Theorem elnp1st2nd 7807

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 7802	. . . . 5
2	1	sseli 3238	. . . 4
3		prop 7806	. . . . . . 7
4		elinp 7805	. . . . . . 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 6382	. . . 4
12	11	ad2antrr 488	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$
13		xp1st 6372	. . . . . . . 8
14	13	elpwid 3685	. . . . . . 7
15		xp2nd 6373	. . . . . . . 8
16	15	elpwid 3685	. . . . . . 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 2311	. 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 2205

wral 2522

wrex 2523

wss 3214

cpw 3674

cop 3697 class class class wbr 4114

cxp 4752

cfv 5357

c1st 6345

c2nd 6346

cnq 7611

cltq 7616

cnp 7622

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 2207 ax-14 2208 ax-ext 2216 ax-coll 4230 ax-sep 4233 ax-pow 4292 ax-pr 4327 ax-un 4559 ax-iinf 4715

This theorem depends on definitions: df-bi 117 df-3an 1007 df-tru 1401 df-nf 1510 df-sb 1812 df-eu 2085 df-mo 2086 df-clab 2221 df-cleq 2227 df-clel 2230 df-nfc 2375 df-ral 2527 df-rex 2528 df-reu 2529 df-rab 2531 df-v 2817 df-sbc 3046 df-csb 3142 df-dif 3216 df-un 3218 df-in 3220 df-ss 3227 df-pw 3676 df-sn 3700 df-pr 3701 df-op 3703 df-uni 3920 df-int 3955 df-iun 3998 df-br 4115 df-opab 4177 df-mpt 4178 df-id 4419 df-iom 4718 df-xp 4760 df-rel 4761 df-cnv 4762 df-co 4763 df-dm 4764 df-rn 4765 df-res 4766 df-ima 4767 df-iota 5317 df-fun 5359 df-fn 5360 df-f 5361 df-f1 5362 df-fo 5363 df-f1o 5364 df-fv 5365 df-1st 6347 df-2nd 6348 df-qs 6786 df-ni 7635 df-nqqs 7679 df-inp 7797

This theorem is referenced by: addclpr 7868 mulclpr 7903 ltexprlempr 7939 recexprlempr 7963 cauappcvgprlemcl 7984 caucvgprlemcl 8007 caucvgprprlemcl 8035