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Theorem dich0 14806

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 4032	. . . . . 6
2		breq2 4033	. . . . . 6
3	1, 2	orbi12d 794	. . . . 5 $\/$ $\/$
4		simpl 109	. . . . 5 $\/$ $/\$ $/\$ $\/$
5		resubcl 8283	. . . . . 6 $/\$
6	5	adantl 277	. . . . 5 $\/$ $/\$ $/\$
7	3, 4, 6	rspcdva 2869	. . . 4 $\/$ $/\$ $/\$ $\/$
8		simprl 529	. . . . . 6 $\/$ $/\$ $/\$
9		simprr 531	. . . . . 6 $\/$ $/\$ $/\$
10	8, 9	suble0d 8555	. . . . 5 $\/$ $/\$ $/\$
11	8, 9	subge0d 8554	. . . . 5 $\/$ $/\$ $/\$
12	10, 11	orbi12d 794	. . . 4 $\/$ $/\$ $/\$ $\/$ $\/$
13	7, 12	mpbid 147	. . 3 $\/$ $/\$ $/\$ $\/$
14	13	ralrimivva 2576	. 2 $\/$ $\/$
15		breq2 4033	. . . . 5
16		breq1 4032	. . . . 5
17	15, 16	orbi12d 794	. . . 4 $\/$ $\/$
18		breq1 4032	. . . . . . 7
19		breq2 4033	. . . . . . 7
20	18, 19	orbi12d 794	. . . . . 6 $\/$ $\/$
21	20	ralbidv 2494	. . . . 5 $\/$ $\/$
22	21	rspccva 2863	. . . 4 $\/$ $/\$ $\/$
23		0red 8020	. . . 4 $\/$ $/\$
24	17, 22, 23	rspcdva 2869	. . 3 $\/$ $/\$ $\/$
25	24	ralrimiva 2567	. 2 $\/$ $\/$
26	14, 25	impbii 126	1 $\/$ $\/$

Colors of variables: wff set class

Syntax hints: $/\$ wa 104

wb 105 $\/$ wo 709

wceq 1364

wcel 2164

wral 2472 class class class wbr 4029 (class class class)co 5918

cr 7871

cc0 7872

cle 8055

cmin 8190

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 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2166 ax-14 2167 ax-ext 2175 ax-sep 4147 ax-pow 4203 ax-pr 4238 ax-un 4464 ax-setind 4569 ax-cnex 7963 ax-resscn 7964 ax-1cn 7965 ax-1re 7966 ax-icn 7967 ax-addcl 7968 ax-addrcl 7969 ax-mulcl 7970 ax-addcom 7972 ax-addass 7974 ax-distr 7976 ax-i2m1 7977 ax-0id 7980 ax-rnegex 7981 ax-cnre 7983 ax-pre-ltadd 7988

This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2045 df-mo 2046 df-clab 2180 df-cleq 2186 df-clel 2189 df-nfc 2325 df-ne 2365 df-nel 2460 df-ral 2477 df-rex 2478 df-reu 2479 df-rab 2481 df-v 2762 df-sbc 2986 df-dif 3155 df-un 3157 df-in 3159 df-ss 3166 df-pw 3603 df-sn 3624 df-pr 3625 df-op 3627 df-uni 3836 df-br 4030 df-opab 4091 df-id 4324 df-xp 4665 df-rel 4666 df-cnv 4667 df-co 4668 df-dm 4669 df-iota 5215 df-fun 5256 df-fv 5262 df-riota 5873 df-ov 5921 df-oprab 5922 df-mpo 5923 df-pnf 8056 df-mnf 8057 df-xr 8058 df-ltxr 8059 df-le 8060 df-sub 8192 df-neg 8193

This theorem is referenced by: ivthdich 14807