dich0 - Intuitionistic Logic Explorer

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Theorem dich0 15379

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 4091	. . . . . 6
2		breq2 4092	. . . . . 6
3	1, 2	orbi12d 800	. . . . 5 $\/$ $\/$
4		simpl 109	. . . . 5 $\/$ $/\$ $/\$ $\/$
5		resubcl 8443	. . . . . 6 $/\$
6	5	adantl 277	. . . . 5 $\/$ $/\$ $/\$
7	3, 4, 6	rspcdva 2915	. . . 4 $\/$ $/\$ $/\$ $\/$
8		simprl 531	. . . . . 6 $\/$ $/\$ $/\$
9		simprr 533	. . . . . 6 $\/$ $/\$ $/\$
10	8, 9	suble0d 8716	. . . . 5 $\/$ $/\$ $/\$
11	8, 9	subge0d 8715	. . . . 5 $\/$ $/\$ $/\$
12	10, 11	orbi12d 800	. . . 4 $\/$ $/\$ $/\$ $\/$ $\/$
13	7, 12	mpbid 147	. . 3 $\/$ $/\$ $/\$ $\/$
14	13	ralrimivva 2614	. 2 $\/$ $\/$
15		breq2 4092	. . . . 5
16		breq1 4091	. . . . 5
17	15, 16	orbi12d 800	. . . 4 $\/$ $\/$
18		breq1 4091	. . . . . . 7
19		breq2 4092	. . . . . . 7
20	18, 19	orbi12d 800	. . . . . 6 $\/$ $\/$
21	20	ralbidv 2532	. . . . 5 $\/$ $\/$
22	21	rspccva 2909	. . . 4 $\/$ $/\$ $\/$
23		0red 8180	. . . 4 $\/$ $/\$
24	17, 22, 23	rspcdva 2915	. . 3 $\/$ $/\$ $\/$
25	24	ralrimiva 2605	. 2 $\/$ $\/$
26	14, 25	impbii 126	1 $\/$ $\/$

Colors of variables: wff set class

Syntax hints: $/\$ wa 104

wb 105 $\/$ wo 715

wceq 1397

wcel 2202

wral 2510 class class class wbr 4088 (class class class)co 6018

cr 8031

cc0 8032

cle 8215

cmin 8350

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 716 ax-5 1495 ax-7 1496 ax-gen 1497 ax-ie1 1541 ax-ie2 1542 ax-8 1552 ax-10 1553 ax-11 1554 ax-i12 1555 ax-bndl 1557 ax-4 1558 ax-17 1574 ax-i9 1578 ax-ial 1582 ax-i5r 1583 ax-13 2204 ax-14 2205 ax-ext 2213 ax-sep 4207 ax-pow 4264 ax-pr 4299 ax-un 4530 ax-setind 4635 ax-cnex 8123 ax-resscn 8124 ax-1cn 8125 ax-1re 8126 ax-icn 8127 ax-addcl 8128 ax-addrcl 8129 ax-mulcl 8130 ax-addcom 8132 ax-addass 8134 ax-distr 8136 ax-i2m1 8137 ax-0id 8140 ax-rnegex 8141 ax-cnre 8143 ax-pre-ltadd 8148

This theorem depends on definitions: df-bi 117 df-3an 1006 df-tru 1400 df-fal 1403 df-nf 1509 df-sb 1811 df-eu 2082 df-mo 2083 df-clab 2218 df-cleq 2224 df-clel 2227 df-nfc 2363 df-ne 2403 df-nel 2498 df-ral 2515 df-rex 2516 df-reu 2517 df-rab 2519 df-v 2804 df-sbc 3032 df-dif 3202 df-un 3204 df-in 3206 df-ss 3213 df-pw 3654 df-sn 3675 df-pr 3676 df-op 3678 df-uni 3894 df-br 4089 df-opab 4151 df-id 4390 df-xp 4731 df-rel 4732 df-cnv 4733 df-co 4734 df-dm 4735 df-iota 5286 df-fun 5328 df-fv 5334 df-riota 5971 df-ov 6021 df-oprab 6022 df-mpo 6023 df-pnf 8216 df-mnf 8217 df-xr 8218 df-ltxr 8219 df-le 8220 df-sub 8352 df-neg 8353

This theorem is referenced by: ivthdich 15380