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Theorem dich0 15291

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 4065	. . . . . 6
2		breq2 4066	. . . . . 6
3	1, 2	orbi12d 797	. . . . 5 $\/$ $\/$
4		simpl 109	. . . . 5 $\/$ $/\$ $/\$ $\/$
5		resubcl 8378	. . . . . 6 $/\$
6	5	adantl 277	. . . . 5 $\/$ $/\$ $/\$
7	3, 4, 6	rspcdva 2892	. . . 4 $\/$ $/\$ $/\$ $\/$
8		simprl 529	. . . . . 6 $\/$ $/\$ $/\$
9		simprr 531	. . . . . 6 $\/$ $/\$ $/\$
10	8, 9	suble0d 8651	. . . . 5 $\/$ $/\$ $/\$
11	8, 9	subge0d 8650	. . . . 5 $\/$ $/\$ $/\$
12	10, 11	orbi12d 797	. . . 4 $\/$ $/\$ $/\$ $\/$ $\/$
13	7, 12	mpbid 147	. . 3 $\/$ $/\$ $/\$ $\/$
14	13	ralrimivva 2592	. 2 $\/$ $\/$
15		breq2 4066	. . . . 5
16		breq1 4065	. . . . 5
17	15, 16	orbi12d 797	. . . 4 $\/$ $\/$
18		breq1 4065	. . . . . . 7
19		breq2 4066	. . . . . . 7
20	18, 19	orbi12d 797	. . . . . 6 $\/$ $\/$
21	20	ralbidv 2510	. . . . 5 $\/$ $\/$
22	21	rspccva 2886	. . . 4 $\/$ $/\$ $\/$
23		0red 8115	. . . 4 $\/$ $/\$
24	17, 22, 23	rspcdva 2892	. . 3 $\/$ $/\$ $\/$
25	24	ralrimiva 2583	. 2 $\/$ $\/$
26	14, 25	impbii 126	1 $\/$ $\/$

Colors of variables: wff set class

Syntax hints: $/\$ wa 104

wb 105 $\/$ wo 712

wceq 1375

wcel 2180

wral 2488 class class class wbr 4062 (class class class)co 5974

cr 7966

cc0 7967

cle 8150

cmin 8285

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 713 ax-5 1473 ax-7 1474 ax-gen 1475 ax-ie1 1519 ax-ie2 1520 ax-8 1530 ax-10 1531 ax-11 1532 ax-i12 1533 ax-bndl 1535 ax-4 1536 ax-17 1552 ax-i9 1556 ax-ial 1560 ax-i5r 1561 ax-13 2182 ax-14 2183 ax-ext 2191 ax-sep 4181 ax-pow 4237 ax-pr 4272 ax-un 4501 ax-setind 4606 ax-cnex 8058 ax-resscn 8059 ax-1cn 8060 ax-1re 8061 ax-icn 8062 ax-addcl 8063 ax-addrcl 8064 ax-mulcl 8065 ax-addcom 8067 ax-addass 8069 ax-distr 8071 ax-i2m1 8072 ax-0id 8075 ax-rnegex 8076 ax-cnre 8078 ax-pre-ltadd 8083

This theorem depends on definitions: df-bi 117 df-3an 985 df-tru 1378 df-fal 1381 df-nf 1487 df-sb 1789 df-eu 2060 df-mo 2061 df-clab 2196 df-cleq 2202 df-clel 2205 df-nfc 2341 df-ne 2381 df-nel 2476 df-ral 2493 df-rex 2494 df-reu 2495 df-rab 2497 df-v 2781 df-sbc 3009 df-dif 3179 df-un 3181 df-in 3183 df-ss 3190 df-pw 3631 df-sn 3652 df-pr 3653 df-op 3655 df-uni 3868 df-br 4063 df-opab 4125 df-id 4361 df-xp 4702 df-rel 4703 df-cnv 4704 df-co 4705 df-dm 4706 df-iota 5254 df-fun 5296 df-fv 5302 df-riota 5927 df-ov 5977 df-oprab 5978 df-mpo 5979 df-pnf 8151 df-mnf 8152 df-xr 8153 df-ltxr 8154 df-le 8155 df-sub 8287 df-neg 8288

This theorem is referenced by: ivthdich 15292