reap0 - Mathbox for Jim Kingdon

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Theorem reap0 14726

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 7956	. . . . . 6
4		breq1 4006	. . . . . . . 8
5		equequ1 1712	. . . . . . . 8
6		breq2 4007	. . . . . . . 8
7	4, 5, 6	3orbi123d 1311	. . . . . . 7 $\/$ $\/$ $\/$ $\/$
8		breq2 4007	. . . . . . . 8
9		eqeq2 2187	. . . . . . . 8
10		breq1 4006	. . . . . . . 8
11	8, 9, 10	3orbi123d 1311	. . . . . . 7 $\/$ $\/$ $\/$ $\/$
12	7, 11	rspc2v 2854	. . . . . 6 $/\$ $\/$ $\/$ $\/$ $\/$
13	2, 3, 12	sylancl 413	. . . . 5 $\/$ $\/$ $/\$ $\/$ $\/$ $\/$ $\/$
14	1, 13	mpd 13	. . . 4 $\/$ $\/$ $/\$ $\/$ $\/$
15		triap 14697	. . . . 5 $/\$ $\/$ $\/$ DECID #
16	2, 3, 15	sylancl 413	. . . 4 $\/$ $\/$ $/\$ $\/$ $\/$ DECID #
17	14, 16	mpbid 147	. . 3 $\/$ $\/$ $/\$ DECID #
18	17	ralrimiva 2550	. 2 $\/$ $\/$ DECID #
19		breq1 4006	. . . . . . 7 # #
20	19	dcbid 838	. . . . . 6 DECID # DECID #
21		simpl 109	. . . . . 6 DECID # $/\$ $/\$ DECID #
22		resubcl 8219	. . . . . . 7 $/\$
23	22	adantl 277	. . . . . 6 DECID # $/\$ $/\$
24	20, 21, 23	rspcdva 2846	. . . . 5 DECID # $/\$ $/\$ DECID #
25		simprl 529	. . . . . . . 8 DECID # $/\$ $/\$
26	25	recnd 7984	. . . . . . 7 DECID # $/\$ $/\$
27		simprr 531	. . . . . . . 8 DECID # $/\$ $/\$
28	27	recnd 7984	. . . . . . 7 DECID # $/\$ $/\$
29		subap0 8598	. . . . . . 7 $/\$ # #
30	26, 28, 29	syl2anc 411	. . . . . 6 DECID # $/\$ $/\$ # #
31	30	dcbid 838	. . . . 5 DECID # $/\$ $/\$ DECID # DECID #
32	24, 31	mpbid 147	. . . 4 DECID # $/\$ $/\$ DECID #
33		triap 14697	. . . . 5 $/\$ $\/$ $\/$ DECID #
34	33	adantl 277	. . . 4 DECID # $/\$ $/\$ $\/$ $\/$ DECID #
35	32, 34	mpbird 167	. . 3 DECID # $/\$ $/\$ $\/$ $\/$
36	35	ralrimivva 2559	. 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 834 $\/$ w3o 977

wceq 1353

wcel 2148

wral 2455 class class class wbr 4003 (class class class)co 5874

cc 7808

cr 7809

cc0 7810

clt 7990

cmin 8126 # cap 8536

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 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4121 ax-pow 4174 ax-pr 4209 ax-un 4433 ax-setind 4536 ax-cnex 7901 ax-resscn 7902 ax-1cn 7903 ax-1re 7904 ax-icn 7905 ax-addcl 7906 ax-addrcl 7907 ax-mulcl 7908 ax-mulrcl 7909 ax-addcom 7910 ax-mulcom 7911 ax-addass 7912 ax-mulass 7913 ax-distr 7914 ax-i2m1 7915 ax-0lt1 7916 ax-1rid 7917 ax-0id 7918 ax-rnegex 7919 ax-precex 7920 ax-cnre 7921 ax-pre-ltirr 7922 ax-pre-lttrn 7924 ax-pre-apti 7925 ax-pre-ltadd 7926 ax-pre-mulgt0 7927

This theorem depends on definitions: df-bi 117 df-dc 835 df-3or 979 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-reu 2462 df-rab 2464 df-v 2739 df-sbc 2963 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-pw 3577 df-sn 3598 df-pr 3599 df-op 3601 df-uni 3810 df-br 4004 df-opab 4065 df-id 4293 df-xp 4632 df-rel 4633 df-cnv 4634 df-co 4635 df-dm 4636 df-iota 5178 df-fun 5218 df-fv 5224 df-riota 5830 df-ov 5877 df-oprab 5878 df-mpo 5879 df-pnf 7992 df-mnf 7993 df-ltxr 7995 df-sub 8128 df-neg 8129 df-reap 8530 df-ap 8537

This theorem is referenced by: dcapnconstALT 14729

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