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Theorem prloc 7394

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 7377	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$
2		simpr3 990	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$ $\/$
3	1, 2	sylbi 120	. . 3 $\/$
4	3	adantr 274	. 2 $/\$ $\/$
5		simpr 109	. 2 $/\$
6		ltrelnq 7268	. . . . . . 7
7	6	brel 4635	. . . . . 6 $/\$
8	7	simpld 111	. . . . 5
9	8	adantl 275	. . . 4 $/\$
10		simpr 109	. . . . . . 7 $/\$ $/\$
11	10	breq1d 3975	. . . . . 6 $/\$ $/\$
12	10	eleq1d 2226	. . . . . . 7 $/\$ $/\$
13	12	orbi1d 781	. . . . . 6 $/\$ $/\$ $\/$ $\/$
14	11, 13	imbi12d 233	. . . . 5 $/\$ $/\$ $\/$ $\/$
15	14	ralbidv 2457	. . . 4 $/\$ $/\$ $\/$ $\/$
16	9, 15	rspcdv 2819	. . 3 $/\$ $\/$ $\/$
17	7	simprd 113	. . . . 5
18	17	adantl 275	. . . 4 $/\$
19		simpr 109	. . . . . 6 $/\$ $/\$
20	19	breq2d 3977	. . . . 5 $/\$ $/\$
21	19	eleq1d 2226	. . . . . 6 $/\$ $/\$
22	21	orbi2d 780	. . . . 5 $/\$ $/\$ $\/$ $\/$
23	20, 22	imbi12d 233	. . . 4 $/\$ $/\$ $\/$ $\/$
24	18, 23	rspcdv 2819	. . 3 $/\$ $\/$ $\/$
25	16, 24	syld 45	. 2 $/\$ $\/$ $\/$
26	4, 5, 25	mp2d 47	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

cop 3563 class class class wbr 3965

cnq 7183

cltq 7188

cnp 7194

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 4134 ax-pr 4168 ax-un 4392 ax-iinf 4545

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 4252 df-iom 4548 df-xp 4589 df-rel 4590 df-cnv 4591 df-co 4592 df-dm 4593 df-rn 4594 df-res 4595 df-ima 4596 df-iota 5132 df-fun 5169 df-fn 5170 df-f 5171 df-f1 5172 df-fo 5173 df-f1o 5174 df-fv 5175 df-qs 6479 df-ni 7207 df-nqqs 7251 df-ltnqqs 7256 df-inp 7369