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Theorem prloc 6589

Description: A Dedekind cut is located. (Contributed by Jim Kingdon, 23-Oct-2019.)

**Assertion**
Ref	Expression
prloc	$/\$ $\/$

Proof of Theorem prloc Dummy variables

are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elinp 6572	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$
2		simpr3 912	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$ $\/$
3	1, 2	sylbi 114	. . 3 $\/$
4	3	adantr 261	. 2 $/\$ $\/$
5		simpr 103	. 2 $/\$
6		ltrelnq 6463	. . . . . . 7
7	6	brel 4392	. . . . . 6 $/\$
8	7	simpld 105	. . . . 5
9	8	adantl 262	. . . 4 $/\$
10		simpr 103	. . . . . . 7 $/\$ $/\$
11	10	breq1d 3774	. . . . . 6 $/\$ $/\$
12	10	eleq1d 2106	. . . . . . 7 $/\$ $/\$
13	12	orbi1d 705	. . . . . 6 $/\$ $/\$ $\/$ $\/$
14	11, 13	imbi12d 223	. . . . 5 $/\$ $/\$ $\/$ $\/$
15	14	ralbidv 2326	. . . 4 $/\$ $/\$ $\/$ $\/$
16	9, 15	rspcdv 2659	. . 3 $/\$ $\/$ $\/$
17	7	simprd 107	. . . . 5
18	17	adantl 262	. . . 4 $/\$
19		simpr 103	. . . . . 6 $/\$ $/\$
20	19	breq2d 3776	. . . . 5 $/\$ $/\$
21	19	eleq1d 2106	. . . . . 6 $/\$ $/\$
22	21	orbi2d 704	. . . . 5 $/\$ $/\$ $\/$ $\/$
23	20, 22	imbi12d 223	. . . 4 $/\$ $/\$ $\/$ $\/$
24	18, 23	rspcdv 2659	. . 3 $/\$ $\/$ $\/$
25	16, 24	syld 40	. 2 $/\$ $\/$ $\/$
26	4, 5, 25	mp2d 41	1 $/\$ $\/$

Colors of variables: wff set class

Syntax hints:

wn 3

wi 4 $/\$ wa 97

wb 98 $\/$ wo 629 $/\$ w3a 885

wceq 1243

wcel 1393

wral 2306

wrex 2307

wss 2917

cop 3378 class class class wbr 3764

cnq 6378

cltq 6383

cnp 6389

This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-13 1404 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-coll 3872 ax-sep 3875 ax-pow 3927 ax-pr 3944 ax-un 4170 ax-iinf 4311

This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ral 2311 df-rex 2312 df-reu 2313 df-rab 2315 df-v 2559 df-sbc 2765 df-csb 2853 df-dif 2920 df-un 2922 df-in 2924 df-ss 2931 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-uni 3581 df-int 3616 df-iun 3659 df-br 3765 df-opab 3819 df-mpt 3820 df-id 4030 df-iom 4314 df-xp 4351 df-rel 4352 df-cnv 4353 df-co 4354 df-dm 4355 df-rn 4356 df-res 4357 df-ima 4358 df-iota 4867 df-fun 4904 df-fn 4905 df-f 4906 df-f1 4907 df-fo 4908 df-f1o 4909 df-fv 4910 df-qs 6112 df-ni 6402 df-nqqs 6446 df-ltnqqs 6451 df-inp 6564