dich0 - Intuitionistic Logic Explorer

	Intuitionistic Logic Explorer		< Previous Next > Nearby theorems
Mirrors > Home > ILE Home > Th. List > dich0		Unicode version

Theorem dich0 15446

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 4096	. . . . . 6
2		breq2 4097	. . . . . 6
3	1, 2	orbi12d 801	. . . . 5 $\/$ $\/$
4		simpl 109	. . . . 5 $\/$ $/\$ $/\$ $\/$
5		resubcl 8485	. . . . . 6 $/\$
6	5	adantl 277	. . . . 5 $\/$ $/\$ $/\$
7	3, 4, 6	rspcdva 2916	. . . 4 $\/$ $/\$ $/\$ $\/$
8		simprl 531	. . . . . 6 $\/$ $/\$ $/\$
9		simprr 533	. . . . . 6 $\/$ $/\$ $/\$
10	8, 9	suble0d 8758	. . . . 5 $\/$ $/\$ $/\$
11	8, 9	subge0d 8757	. . . . 5 $\/$ $/\$ $/\$
12	10, 11	orbi12d 801	. . . 4 $\/$ $/\$ $/\$ $\/$ $\/$
13	7, 12	mpbid 147	. . 3 $\/$ $/\$ $/\$ $\/$
14	13	ralrimivva 2615	. 2 $\/$ $\/$
15		breq2 4097	. . . . 5
16		breq1 4096	. . . . 5
17	15, 16	orbi12d 801	. . . 4 $\/$ $\/$
18		breq1 4096	. . . . . . 7
19		breq2 4097	. . . . . . 7
20	18, 19	orbi12d 801	. . . . . 6 $\/$ $\/$
21	20	ralbidv 2533	. . . . 5 $\/$ $\/$
22	21	rspccva 2910	. . . 4 $\/$ $/\$ $\/$
23		0red 8223	. . . 4 $\/$ $/\$
24	17, 22, 23	rspcdva 2916	. . 3 $\/$ $/\$ $\/$
25	24	ralrimiva 2606	. 2 $\/$ $\/$
26	14, 25	impbii 126	1 $\/$ $\/$

Colors of variables: wff set class

Syntax hints: $/\$ wa 104

wb 105 $\/$ wo 716

wceq 1398

wcel 2202

wral 2511 class class class wbr 4093 (class class class)co 6028

cr 8074

cc0 8075

cle 8257

cmin 8392

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 717 ax-5 1496 ax-7 1497 ax-gen 1498 ax-ie1 1542 ax-ie2 1543 ax-8 1553 ax-10 1554 ax-11 1555 ax-i12 1556 ax-bndl 1558 ax-4 1559 ax-17 1575 ax-i9 1579 ax-ial 1583 ax-i5r 1584 ax-13 2204 ax-14 2205 ax-ext 2213 ax-sep 4212 ax-pow 4270 ax-pr 4305 ax-un 4536 ax-setind 4641 ax-cnex 8166 ax-resscn 8167 ax-1cn 8168 ax-1re 8169 ax-icn 8170 ax-addcl 8171 ax-addrcl 8172 ax-mulcl 8173 ax-addcom 8175 ax-addass 8177 ax-distr 8179 ax-i2m1 8180 ax-0id 8183 ax-rnegex 8184 ax-cnre 8186 ax-pre-ltadd 8191

This theorem depends on definitions: df-bi 117 df-3an 1007 df-tru 1401 df-fal 1404 df-nf 1510 df-sb 1811 df-eu 2082 df-mo 2083 df-clab 2218 df-cleq 2224 df-clel 2227 df-nfc 2364 df-ne 2404 df-nel 2499 df-ral 2516 df-rex 2517 df-reu 2518 df-rab 2520 df-v 2805 df-sbc 3033 df-dif 3203 df-un 3205 df-in 3207 df-ss 3214 df-pw 3658 df-sn 3679 df-pr 3680 df-op 3682 df-uni 3899 df-br 4094 df-opab 4156 df-id 4396 df-xp 4737 df-rel 4738 df-cnv 4739 df-co 4740 df-dm 4741 df-iota 5293 df-fun 5335 df-fv 5341 df-riota 5981 df-ov 6031 df-oprab 6032 df-mpo 6033 df-pnf 8258 df-mnf 8259 df-xr 8260 df-ltxr 8261 df-le 8262 df-sub 8394 df-neg 8395

This theorem is referenced by: ivthdich 15447