reap0 - Mathbox for Jim Kingdon

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Theorem reap0 16426

Description: Real number trichotomy is equivalent to decidability of apartness from zero. (Contributed by Jim Kingdon, 27-Jul-2024.)

**Assertion**
Ref	Expression
reap0	$\/$ $\/$ DECID #

Distinct variable group:

**Proof of Theorem reap0**
Step	Hyp	Ref	Expression
1		simpl 109	. . . . 5 $\/$ $\/$ $/\$ $\/$ $\/$
2		simpr 110	. . . . . 6 $\/$ $\/$ $/\$
3		0re 8146	. . . . . 6
4		breq1 4086	. . . . . . . 8
5		equequ1 1758	. . . . . . . 8
6		breq2 4087	. . . . . . . 8
7	4, 5, 6	3orbi123d 1345	. . . . . . 7 $\/$ $\/$ $\/$ $\/$
8		breq2 4087	. . . . . . . 8
9		eqeq2 2239	. . . . . . . 8
10		breq1 4086	. . . . . . . 8
11	8, 9, 10	3orbi123d 1345	. . . . . . 7 $\/$ $\/$ $\/$ $\/$
12	7, 11	rspc2v 2920	. . . . . 6 $/\$ $\/$ $\/$ $\/$ $\/$
13	2, 3, 12	sylancl 413	. . . . 5 $\/$ $\/$ $/\$ $\/$ $\/$ $\/$ $\/$
14	1, 13	mpd 13	. . . 4 $\/$ $\/$ $/\$ $\/$ $\/$
15		triap 16397	. . . . 5 $/\$ $\/$ $\/$ DECID #
16	2, 3, 15	sylancl 413	. . . 4 $\/$ $\/$ $/\$ $\/$ $\/$ DECID #
17	14, 16	mpbid 147	. . 3 $\/$ $\/$ $/\$ DECID #
18	17	ralrimiva 2603	. 2 $\/$ $\/$ DECID #
19		breq1 4086	. . . . . . 7 # #
20	19	dcbid 843	. . . . . 6 DECID # DECID #
21		simpl 109	. . . . . 6 DECID # $/\$ $/\$ DECID #
22		resubcl 8410	. . . . . . 7 $/\$
23	22	adantl 277	. . . . . 6 DECID # $/\$ $/\$
24	20, 21, 23	rspcdva 2912	. . . . 5 DECID # $/\$ $/\$ DECID #
25		simprl 529	. . . . . . . 8 DECID # $/\$ $/\$
26	25	recnd 8175	. . . . . . 7 DECID # $/\$ $/\$
27		simprr 531	. . . . . . . 8 DECID # $/\$ $/\$
28	27	recnd 8175	. . . . . . 7 DECID # $/\$ $/\$
29		subap0 8790	. . . . . . 7 $/\$ # #
30	26, 28, 29	syl2anc 411	. . . . . 6 DECID # $/\$ $/\$ # #
31	30	dcbid 843	. . . . 5 DECID # $/\$ $/\$ DECID # DECID #
32	24, 31	mpbid 147	. . . 4 DECID # $/\$ $/\$ DECID #
33		triap 16397	. . . . 5 $/\$ $\/$ $\/$ DECID #
34	33	adantl 277	. . . 4 DECID # $/\$ $/\$ $\/$ $\/$ DECID #
35	32, 34	mpbird 167	. . 3 DECID # $/\$ $/\$ $\/$ $\/$
36	35	ralrimivva 2612	. 2 DECID # $\/$ $\/$
37	18, 36	impbii 126	1 $\/$ $\/$ DECID #

Colors of variables: wff set class

Syntax hints:

wi 4 $/\$ wa 104

wb 105 DECID wdc 839 $\/$ w3o 1001

wceq 1395

wcel 2200

wral 2508 class class class wbr 4083 (class class class)co 6001

cc 7997

cr 7998

cc0 7999

clt 8181

cmin 8317 # cap 8728

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 617 ax-in2 618 ax-io 714 ax-5 1493 ax-7 1494 ax-gen 1495 ax-ie1 1539 ax-ie2 1540 ax-8 1550 ax-10 1551 ax-11 1552 ax-i12 1553 ax-bndl 1555 ax-4 1556 ax-17 1572 ax-i9 1576 ax-ial 1580 ax-i5r 1581 ax-13 2202 ax-14 2203 ax-ext 2211 ax-sep 4202 ax-pow 4258 ax-pr 4293 ax-un 4524 ax-setind 4629 ax-cnex 8090 ax-resscn 8091 ax-1cn 8092 ax-1re 8093 ax-icn 8094 ax-addcl 8095 ax-addrcl 8096 ax-mulcl 8097 ax-mulrcl 8098 ax-addcom 8099 ax-mulcom 8100 ax-addass 8101 ax-mulass 8102 ax-distr 8103 ax-i2m1 8104 ax-0lt1 8105 ax-1rid 8106 ax-0id 8107 ax-rnegex 8108 ax-precex 8109 ax-cnre 8110 ax-pre-ltirr 8111 ax-pre-lttrn 8113 ax-pre-apti 8114 ax-pre-ltadd 8115 ax-pre-mulgt0 8116

This theorem depends on definitions: df-bi 117 df-dc 840 df-3or 1003 df-3an 1004 df-tru 1398 df-fal 1401 df-nf 1507 df-sb 1809 df-eu 2080 df-mo 2081 df-clab 2216 df-cleq 2222 df-clel 2225 df-nfc 2361 df-ne 2401 df-nel 2496 df-ral 2513 df-rex 2514 df-reu 2515 df-rab 2517 df-v 2801 df-sbc 3029 df-dif 3199 df-un 3201 df-in 3203 df-ss 3210 df-pw 3651 df-sn 3672 df-pr 3673 df-op 3675 df-uni 3889 df-br 4084 df-opab 4146 df-id 4384 df-xp 4725 df-rel 4726 df-cnv 4727 df-co 4728 df-dm 4729 df-iota 5278 df-fun 5320 df-fv 5326 df-riota 5954 df-ov 6004 df-oprab 6005 df-mpo 6006 df-pnf 8183 df-mnf 8184 df-ltxr 8186 df-sub 8319 df-neg 8320 df-reap 8722 df-ap 8729

This theorem is referenced by: dcapnconstALT 16430

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