reap0 - Mathbox for Jim Kingdon

	Mathbox for Jim Kingdon		< Previous Next > Nearby theorems
Mirrors > Home > ILE Home > Th. List > Mathboxes > reap0		Unicode version

Theorem reap0 16860

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 8276	. . . . . 6
4		breq1 4114	. . . . . . . 8
5		equequ1 1760	. . . . . . . 8
6		breq2 4115	. . . . . . . 8
7	4, 5, 6	3orbi123d 1348	. . . . . . 7 $\/$ $\/$ $\/$ $\/$
8		breq2 4115	. . . . . . . 8
9		eqeq2 2244	. . . . . . . 8
10		breq1 4114	. . . . . . . 8
11	8, 9, 10	3orbi123d 1348	. . . . . . 7 $\/$ $\/$ $\/$ $\/$
12	7, 11	rspc2v 2936	. . . . . 6 $/\$ $\/$ $\/$ $\/$ $\/$
13	2, 3, 12	sylancl 413	. . . . 5 $\/$ $\/$ $/\$ $\/$ $\/$ $\/$ $\/$
14	1, 13	mpd 13	. . . 4 $\/$ $\/$ $/\$ $\/$ $\/$
15		triap 16830	. . . . 5 $/\$ $\/$ $\/$ DECID #
16	2, 3, 15	sylancl 413	. . . 4 $\/$ $\/$ $/\$ $\/$ $\/$ DECID #
17	14, 16	mpbid 147	. . 3 $\/$ $\/$ $/\$ DECID #
18	17	ralrimiva 2617	. 2 $\/$ $\/$ DECID #
19		breq1 4114	. . . . . . 7 # #
20	19	dcbid 846	. . . . . 6 DECID # DECID #
21		simpl 109	. . . . . 6 DECID # $/\$ $/\$ DECID #
22		resubcl 8539	. . . . . . 7 $/\$
23	22	adantl 277	. . . . . 6 DECID # $/\$ $/\$
24	20, 21, 23	rspcdva 2928	. . . . 5 DECID # $/\$ $/\$ DECID #
25		simprl 531	. . . . . . . 8 DECID # $/\$ $/\$
26	25	recnd 8304	. . . . . . 7 DECID # $/\$ $/\$
27		simprr 533	. . . . . . . 8 DECID # $/\$ $/\$
28	27	recnd 8304	. . . . . . 7 DECID # $/\$ $/\$
29		subap0 8919	. . . . . . 7 $/\$ # #
30	26, 28, 29	syl2anc 411	. . . . . 6 DECID # $/\$ $/\$ # #
31	30	dcbid 846	. . . . 5 DECID # $/\$ $/\$ DECID # DECID #
32	24, 31	mpbid 147	. . . 4 DECID # $/\$ $/\$ DECID #
33		triap 16830	. . . . 5 $/\$ $\/$ $\/$ DECID #
34	33	adantl 277	. . . 4 DECID # $/\$ $/\$ $\/$ $\/$ DECID #
35	32, 34	mpbird 167	. . 3 DECID # $/\$ $/\$ $\/$ $\/$
36	35	ralrimivva 2626	. 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 842 $\/$ w3o 1004

wceq 1398

wcel 2205

wral 2522 class class class wbr 4111 (class class class)co 6052

cc 8127

cr 8128

cc0 8129

clt 8310

cmin 8446 # cap 8857

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 619 ax-in2 620 ax-io 717 ax-5 1496 ax-7 1497 ax-gen 1498 ax-ie1 1542 ax-ie2 1543 ax-8 1553 ax-10 1554 ax-11 1555 ax-i12 1556 ax-bndl 1558 ax-4 1559 ax-17 1575 ax-i9 1579 ax-ial 1583 ax-i5r 1584 ax-13 2207 ax-14 2208 ax-ext 2216 ax-sep 4230 ax-pow 4289 ax-pr 4324 ax-un 4556 ax-setind 4661 ax-cnex 8220 ax-resscn 8221 ax-1cn 8222 ax-1re 8223 ax-icn 8224 ax-addcl 8225 ax-addrcl 8226 ax-mulcl 8227 ax-mulrcl 8228 ax-addcom 8229 ax-mulcom 8230 ax-addass 8231 ax-mulass 8232 ax-distr 8233 ax-i2m1 8234 ax-0lt1 8235 ax-1rid 8236 ax-0id 8237 ax-rnegex 8238 ax-precex 8239 ax-cnre 8240 ax-pre-ltirr 8241 ax-pre-lttrn 8243 ax-pre-apti 8244 ax-pre-ltadd 8245 ax-pre-mulgt0 8246

This theorem depends on definitions: df-bi 117 df-dc 843 df-3or 1006 df-3an 1007 df-tru 1401 df-fal 1404 df-nf 1510 df-sb 1812 df-eu 2085 df-mo 2086 df-clab 2221 df-cleq 2227 df-clel 2230 df-nfc 2375 df-ne 2415 df-nel 2510 df-ral 2527 df-rex 2528 df-reu 2529 df-rab 2531 df-v 2817 df-sbc 3045 df-dif 3215 df-un 3217 df-in 3219 df-ss 3226 df-pw 3673 df-sn 3697 df-pr 3698 df-op 3700 df-uni 3917 df-br 4112 df-opab 4174 df-id 4416 df-xp 4757 df-rel 4758 df-cnv 4759 df-co 4760 df-dm 4761 df-iota 5314 df-fun 5356 df-fv 5362 df-riota 6005 df-ov 6055 df-oprab 6056 df-mpo 6057 df-pnf 8312 df-mnf 8313 df-ltxr 8315 df-sub 8448 df-neg 8449 df-reap 8851 df-ap 8858

This theorem is referenced by: dcapnconstALT 16865

W3C validator