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Theorem prloc 7292

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 7275	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$
2		simpr3 989	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$ $\/$
3	1, 2	sylbi 120	. . 3 $\/$
4	3	adantr 274	. 2 $/\$ $\/$
5		simpr 109	. 2 $/\$
6		ltrelnq 7166	. . . . . . 7
7	6	brel 4586	. . . . . 6 $/\$
8	7	simpld 111	. . . . 5
9	8	adantl 275	. . . 4 $/\$
10		simpr 109	. . . . . . 7 $/\$ $/\$
11	10	breq1d 3934	. . . . . 6 $/\$ $/\$
12	10	eleq1d 2206	. . . . . . 7 $/\$ $/\$
13	12	orbi1d 780	. . . . . 6 $/\$ $/\$ $\/$ $\/$
14	11, 13	imbi12d 233	. . . . 5 $/\$ $/\$ $\/$ $\/$
15	14	ralbidv 2435	. . . 4 $/\$ $/\$ $\/$ $\/$
16	9, 15	rspcdv 2787	. . 3 $/\$ $\/$ $\/$
17	7	simprd 113	. . . . 5
18	17	adantl 275	. . . 4 $/\$
19		simpr 109	. . . . . 6 $/\$ $/\$
20	19	breq2d 3936	. . . . 5 $/\$ $/\$
21	19	eleq1d 2206	. . . . . 6 $/\$ $/\$
22	21	orbi2d 779	. . . . 5 $/\$ $/\$ $\/$ $\/$
23	20, 22	imbi12d 233	. . . 4 $/\$ $/\$ $\/$ $\/$
24	18, 23	rspcdv 2787	. . 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 697 $/\$ w3a 962

wceq 1331

wcel 1480

wral 2414

wrex 2415

wss 3066

cop 3525 class class class wbr 3924

cnq 7081

cltq 7086

cnp 7092

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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-coll 4038 ax-sep 4041 ax-pow 4093 ax-pr 4126 ax-un 4350 ax-iinf 4497

This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ral 2419 df-rex 2420 df-reu 2421 df-rab 2423 df-v 2683 df-sbc 2905 df-csb 2999 df-dif 3068 df-un 3070 df-in 3072 df-ss 3079 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-int 3767 df-iun 3810 df-br 3925 df-opab 3985 df-mpt 3986 df-id 4210 df-iom 4500 df-xp 4540 df-rel 4541 df-cnv 4542 df-co 4543 df-dm 4544 df-rn 4545 df-res 4546 df-ima 4547 df-iota 5083 df-fun 5120 df-fn 5121 df-f 5122 df-f1 5123 df-fo 5124 df-f1o 5125 df-fv 5126 df-qs 6428 df-ni 7105 df-nqqs 7149 df-ltnqqs 7154 df-inp 7267