reap0 - Mathbox for Jim Kingdon

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Theorem reap0 13937

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 108	. . . . 5 $\/$ $\/$ $/\$ $\/$ $\/$
2		simpr 109	. . . . . 6 $\/$ $\/$ $/\$
3		0re 7899	. . . . . 6
4		breq1 3985	. . . . . . . 8
5		equequ1 1700	. . . . . . . 8
6		breq2 3986	. . . . . . . 8
7	4, 5, 6	3orbi123d 1301	. . . . . . 7 $\/$ $\/$ $\/$ $\/$
8		breq2 3986	. . . . . . . 8
9		eqeq2 2175	. . . . . . . 8
10		breq1 3985	. . . . . . . 8
11	8, 9, 10	3orbi123d 1301	. . . . . . 7 $\/$ $\/$ $\/$ $\/$
12	7, 11	rspc2v 2843	. . . . . 6 $/\$ $\/$ $\/$ $\/$ $\/$
13	2, 3, 12	sylancl 410	. . . . 5 $\/$ $\/$ $/\$ $\/$ $\/$ $\/$ $\/$
14	1, 13	mpd 13	. . . 4 $\/$ $\/$ $/\$ $\/$ $\/$
15		triap 13908	. . . . 5 $/\$ $\/$ $\/$ DECID #
16	2, 3, 15	sylancl 410	. . . 4 $\/$ $\/$ $/\$ $\/$ $\/$ DECID #
17	14, 16	mpbid 146	. . 3 $\/$ $\/$ $/\$ DECID #
18	17	ralrimiva 2539	. 2 $\/$ $\/$ DECID #
19		breq1 3985	. . . . . . 7 # #
20	19	dcbid 828	. . . . . 6 DECID # DECID #
21		simpl 108	. . . . . 6 DECID # $/\$ $/\$ DECID #
22		resubcl 8162	. . . . . . 7 $/\$
23	22	adantl 275	. . . . . 6 DECID # $/\$ $/\$
24	20, 21, 23	rspcdva 2835	. . . . 5 DECID # $/\$ $/\$ DECID #
25		simprl 521	. . . . . . . 8 DECID # $/\$ $/\$
26	25	recnd 7927	. . . . . . 7 DECID # $/\$ $/\$
27		simprr 522	. . . . . . . 8 DECID # $/\$ $/\$
28	27	recnd 7927	. . . . . . 7 DECID # $/\$ $/\$
29		subap0 8541	. . . . . . 7 $/\$ # #
30	26, 28, 29	syl2anc 409	. . . . . 6 DECID # $/\$ $/\$ # #
31	30	dcbid 828	. . . . 5 DECID # $/\$ $/\$ DECID # DECID #
32	24, 31	mpbid 146	. . . 4 DECID # $/\$ $/\$ DECID #
33		triap 13908	. . . . 5 $/\$ $\/$ $\/$ DECID #
34	33	adantl 275	. . . 4 DECID # $/\$ $/\$ $\/$ $\/$ DECID #
35	32, 34	mpbird 166	. . 3 DECID # $/\$ $/\$ $\/$ $\/$
36	35	ralrimivva 2548	. 2 DECID # $\/$ $\/$
37	18, 36	impbii 125	1 $\/$ $\/$ DECID #

Colors of variables: wff set class

Syntax hints:

wi 4 $/\$ wa 103

wb 104 DECID wdc 824 $\/$ w3o 967

wceq 1343

wcel 2136

wral 2444 class class class wbr 3982 (class class class)co 5842

cc 7751

cr 7752

cc0 7753

clt 7933

cmin 8069 # cap 8479

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 1435 ax-7 1436 ax-gen 1437 ax-ie1 1481 ax-ie2 1482 ax-8 1492 ax-10 1493 ax-11 1494 ax-i12 1495 ax-bndl 1497 ax-4 1498 ax-17 1514 ax-i9 1518 ax-ial 1522 ax-i5r 1523 ax-13 2138 ax-14 2139 ax-ext 2147 ax-sep 4100 ax-pow 4153 ax-pr 4187 ax-un 4411 ax-setind 4514 ax-cnex 7844 ax-resscn 7845 ax-1cn 7846 ax-1re 7847 ax-icn 7848 ax-addcl 7849 ax-addrcl 7850 ax-mulcl 7851 ax-mulrcl 7852 ax-addcom 7853 ax-mulcom 7854 ax-addass 7855 ax-mulass 7856 ax-distr 7857 ax-i2m1 7858 ax-0lt1 7859 ax-1rid 7860 ax-0id 7861 ax-rnegex 7862 ax-precex 7863 ax-cnre 7864 ax-pre-ltirr 7865 ax-pre-lttrn 7867 ax-pre-apti 7868 ax-pre-ltadd 7869 ax-pre-mulgt0 7870

This theorem depends on definitions: df-bi 116 df-dc 825 df-3or 969 df-3an 970 df-tru 1346 df-fal 1349 df-nf 1449 df-sb 1751 df-eu 2017 df-mo 2018 df-clab 2152 df-cleq 2158 df-clel 2161 df-nfc 2297 df-ne 2337 df-nel 2432 df-ral 2449 df-rex 2450 df-reu 2451 df-rab 2453 df-v 2728 df-sbc 2952 df-dif 3118 df-un 3120 df-in 3122 df-ss 3129 df-pw 3561 df-sn 3582 df-pr 3583 df-op 3585 df-uni 3790 df-br 3983 df-opab 4044 df-id 4271 df-xp 4610 df-rel 4611 df-cnv 4612 df-co 4613 df-dm 4614 df-iota 5153 df-fun 5190 df-fv 5196 df-riota 5798 df-ov 5845 df-oprab 5846 df-mpo 5847 df-pnf 7935 df-mnf 7936 df-ltxr 7938 df-sub 8071 df-neg 8072 df-reap 8473 df-ap 8480

This theorem is referenced by: dcapnconstALT 13940

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