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Theorem dich0 15369

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 4089	. . . . . 6
2		breq2 4090	. . . . . 6
3	1, 2	orbi12d 798	. . . . 5 $\/$ $\/$
4		simpl 109	. . . . 5 $\/$ $/\$ $/\$ $\/$
5		resubcl 8436	. . . . . 6 $/\$
6	5	adantl 277	. . . . 5 $\/$ $/\$ $/\$
7	3, 4, 6	rspcdva 2913	. . . 4 $\/$ $/\$ $/\$ $\/$
8		simprl 529	. . . . . 6 $\/$ $/\$ $/\$
9		simprr 531	. . . . . 6 $\/$ $/\$ $/\$
10	8, 9	suble0d 8709	. . . . 5 $\/$ $/\$ $/\$
11	8, 9	subge0d 8708	. . . . 5 $\/$ $/\$ $/\$
12	10, 11	orbi12d 798	. . . 4 $\/$ $/\$ $/\$ $\/$ $\/$
13	7, 12	mpbid 147	. . 3 $\/$ $/\$ $/\$ $\/$
14	13	ralrimivva 2612	. 2 $\/$ $\/$
15		breq2 4090	. . . . 5
16		breq1 4089	. . . . 5
17	15, 16	orbi12d 798	. . . 4 $\/$ $\/$
18		breq1 4089	. . . . . . 7
19		breq2 4090	. . . . . . 7
20	18, 19	orbi12d 798	. . . . . 6 $\/$ $\/$
21	20	ralbidv 2530	. . . . 5 $\/$ $\/$
22	21	rspccva 2907	. . . 4 $\/$ $/\$ $\/$
23		0red 8173	. . . 4 $\/$ $/\$
24	17, 22, 23	rspcdva 2913	. . 3 $\/$ $/\$ $\/$
25	24	ralrimiva 2603	. 2 $\/$ $\/$
26	14, 25	impbii 126	1 $\/$ $\/$

Colors of variables: wff set class

Syntax hints: $/\$ wa 104

wb 105 $\/$ wo 713

wceq 1395

wcel 2200

wral 2508 class class class wbr 4086 (class class class)co 6013

cr 8024

cc0 8025

cle 8208

cmin 8343

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 617 ax-in2 618 ax-io 714 ax-5 1493 ax-7 1494 ax-gen 1495 ax-ie1 1539 ax-ie2 1540 ax-8 1550 ax-10 1551 ax-11 1552 ax-i12 1553 ax-bndl 1555 ax-4 1556 ax-17 1572 ax-i9 1576 ax-ial 1580 ax-i5r 1581 ax-13 2202 ax-14 2203 ax-ext 2211 ax-sep 4205 ax-pow 4262 ax-pr 4297 ax-un 4528 ax-setind 4633 ax-cnex 8116 ax-resscn 8117 ax-1cn 8118 ax-1re 8119 ax-icn 8120 ax-addcl 8121 ax-addrcl 8122 ax-mulcl 8123 ax-addcom 8125 ax-addass 8127 ax-distr 8129 ax-i2m1 8130 ax-0id 8133 ax-rnegex 8134 ax-cnre 8136 ax-pre-ltadd 8141

This theorem depends on definitions: df-bi 117 df-3an 1004 df-tru 1398 df-fal 1401 df-nf 1507 df-sb 1809 df-eu 2080 df-mo 2081 df-clab 2216 df-cleq 2222 df-clel 2225 df-nfc 2361 df-ne 2401 df-nel 2496 df-ral 2513 df-rex 2514 df-reu 2515 df-rab 2517 df-v 2802 df-sbc 3030 df-dif 3200 df-un 3202 df-in 3204 df-ss 3211 df-pw 3652 df-sn 3673 df-pr 3674 df-op 3676 df-uni 3892 df-br 4087 df-opab 4149 df-id 4388 df-xp 4729 df-rel 4730 df-cnv 4731 df-co 4732 df-dm 4733 df-iota 5284 df-fun 5326 df-fv 5332 df-riota 5966 df-ov 6016 df-oprab 6017 df-mpo 6018 df-pnf 8209 df-mnf 8210 df-xr 8211 df-ltxr 8212 df-le 8213 df-sub 8345 df-neg 8346

This theorem is referenced by: ivthdich 15370