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Theorem dich0 14972

Description: Real number dichotomy stated in terms of two real numbers or a real number and zero. (Contributed by Jim Kingdon, 22-Jul-2025.)

**Assertion**
Ref	Expression
dich0	$\/$ $\/$

**Proof of Theorem dich0**
Step	Hyp	Ref	Expression
1		breq1 4037	. . . . . 6
2		breq2 4038	. . . . . 6
3	1, 2	orbi12d 794	. . . . 5 $\/$ $\/$
4		simpl 109	. . . . 5 $\/$ $/\$ $/\$ $\/$
5		resubcl 8307	. . . . . 6 $/\$
6	5	adantl 277	. . . . 5 $\/$ $/\$ $/\$
7	3, 4, 6	rspcdva 2873	. . . 4 $\/$ $/\$ $/\$ $\/$
8		simprl 529	. . . . . 6 $\/$ $/\$ $/\$
9		simprr 531	. . . . . 6 $\/$ $/\$ $/\$
10	8, 9	suble0d 8580	. . . . 5 $\/$ $/\$ $/\$
11	8, 9	subge0d 8579	. . . . 5 $\/$ $/\$ $/\$
12	10, 11	orbi12d 794	. . . 4 $\/$ $/\$ $/\$ $\/$ $\/$
13	7, 12	mpbid 147	. . 3 $\/$ $/\$ $/\$ $\/$
14	13	ralrimivva 2579	. 2 $\/$ $\/$
15		breq2 4038	. . . . 5
16		breq1 4037	. . . . 5
17	15, 16	orbi12d 794	. . . 4 $\/$ $\/$
18		breq1 4037	. . . . . . 7
19		breq2 4038	. . . . . . 7
20	18, 19	orbi12d 794	. . . . . 6 $\/$ $\/$
21	20	ralbidv 2497	. . . . 5 $\/$ $\/$
22	21	rspccva 2867	. . . 4 $\/$ $/\$ $\/$
23		0red 8044	. . . 4 $\/$ $/\$
24	17, 22, 23	rspcdva 2873	. . 3 $\/$ $/\$ $\/$
25	24	ralrimiva 2570	. 2 $\/$ $\/$
26	14, 25	impbii 126	1 $\/$ $\/$

Colors of variables: wff set class

Syntax hints: $/\$ wa 104

wb 105 $\/$ wo 709

wceq 1364

wcel 2167

wral 2475 class class class wbr 4034 (class class class)co 5925

cr 7895

cc0 7896

cle 8079

cmin 8214

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 615 ax-in2 616 ax-io 710 ax-5 1461 ax-7 1462 ax-gen 1463 ax-ie1 1507 ax-ie2 1508 ax-8 1518 ax-10 1519 ax-11 1520 ax-i12 1521 ax-bndl 1523 ax-4 1524 ax-17 1540 ax-i9 1544 ax-ial 1548 ax-i5r 1549 ax-13 2169 ax-14 2170 ax-ext 2178 ax-sep 4152 ax-pow 4208 ax-pr 4243 ax-un 4469 ax-setind 4574 ax-cnex 7987 ax-resscn 7988 ax-1cn 7989 ax-1re 7990 ax-icn 7991 ax-addcl 7992 ax-addrcl 7993 ax-mulcl 7994 ax-addcom 7996 ax-addass 7998 ax-distr 8000 ax-i2m1 8001 ax-0id 8004 ax-rnegex 8005 ax-cnre 8007 ax-pre-ltadd 8012

This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1475 df-sb 1777 df-eu 2048 df-mo 2049 df-clab 2183 df-cleq 2189 df-clel 2192 df-nfc 2328 df-ne 2368 df-nel 2463 df-ral 2480 df-rex 2481 df-reu 2482 df-rab 2484 df-v 2765 df-sbc 2990 df-dif 3159 df-un 3161 df-in 3163 df-ss 3170 df-pw 3608 df-sn 3629 df-pr 3630 df-op 3632 df-uni 3841 df-br 4035 df-opab 4096 df-id 4329 df-xp 4670 df-rel 4671 df-cnv 4672 df-co 4673 df-dm 4674 df-iota 5220 df-fun 5261 df-fv 5267 df-riota 5880 df-ov 5928 df-oprab 5929 df-mpo 5930 df-pnf 8080 df-mnf 8081 df-xr 8082 df-ltxr 8083 df-le 8084 df-sub 8216 df-neg 8217

This theorem is referenced by: ivthdich 14973