blss2 - Intuitionistic Logic Explorer

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Theorem blss2 14879

Description: One ball is contained in another if the center-to-center distance is less than the difference of the radii. (Contributed by Mario Carneiro, 15-Jan-2014.) (Revised by Mario Carneiro, 23-Aug-2015.)

**Assertion**
Ref	Expression
blss2	$/\$ $/\$ $/\$ $/\$ $/\$

**Proof of Theorem blss2**
Step	Hyp	Ref	Expression
1		simpl1 1003	. 2 $/\$ $/\$ $/\$ $/\$ $/\$
2		simpl2 1004	. 2 $/\$ $/\$ $/\$ $/\$ $/\$
3		simpl3 1005	. 2 $/\$ $/\$ $/\$ $/\$ $/\$
4		simpr1 1006	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$
5	4	rexrd 8122	. 2 $/\$ $/\$ $/\$ $/\$ $/\$
6		simpr2 1007	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$
7	6	rexrd 8122	. 2 $/\$ $/\$ $/\$ $/\$ $/\$
8	6, 4	resubcld 8453	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$
9		simpr3 1008	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$
10		xmetlecl 14839	. . 3 $/\$ $/\$ $/\$ $/\$
11	1, 2, 3, 8, 9, 10	syl122anc 1259	. 2 $/\$ $/\$ $/\$ $/\$ $/\$
12		rexsub 9975	. . . 4 $/\$
13	6, 4, 12	syl2anc 411	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$
14	9, 13	breqtrrd 4072	. 2 $/\$ $/\$ $/\$ $/\$ $/\$
15	1, 2, 3, 5, 7, 11, 14	xblss2 14877	1 $/\$ $/\$ $/\$ $/\$ $/\$

Colors of variables: wff set class

Syntax hints:

wi 4 $/\$ wa 104 $/\$ w3a 981

wceq 1373

wcel 2176

wss 3166 class class class wbr 4044

cfv 5271 (class class class)co 5944

cr 7924

cle 8108

cmin 8243

cxne 9891

cxad 9892

cxmet 14298

cbl 14300

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 711 ax-5 1470 ax-7 1471 ax-gen 1472 ax-ie1 1516 ax-ie2 1517 ax-8 1527 ax-10 1528 ax-11 1529 ax-i12 1530 ax-bndl 1532 ax-4 1533 ax-17 1549 ax-i9 1553 ax-ial 1557 ax-i5r 1558 ax-13 2178 ax-14 2179 ax-ext 2187 ax-sep 4162 ax-pow 4218 ax-pr 4253 ax-un 4480 ax-setind 4585 ax-cnex 8016 ax-resscn 8017 ax-1cn 8018 ax-1re 8019 ax-icn 8020 ax-addcl 8021 ax-addrcl 8022 ax-mulcl 8023 ax-mulrcl 8024 ax-addcom 8025 ax-mulcom 8026 ax-addass 8027 ax-mulass 8028 ax-distr 8029 ax-i2m1 8030 ax-0lt1 8031 ax-1rid 8032 ax-0id 8033 ax-rnegex 8034 ax-precex 8035 ax-cnre 8036 ax-pre-ltirr 8037 ax-pre-ltwlin 8038 ax-pre-lttrn 8039 ax-pre-apti 8040 ax-pre-ltadd 8041 ax-pre-mulgt0 8042

This theorem depends on definitions: df-bi 117 df-stab 833 df-dc 837 df-3or 982 df-3an 983 df-tru 1376 df-fal 1379 df-nf 1484 df-sb 1786 df-eu 2057 df-mo 2058 df-clab 2192 df-cleq 2198 df-clel 2201 df-nfc 2337 df-ne 2377 df-nel 2472 df-ral 2489 df-rex 2490 df-reu 2491 df-rab 2493 df-v 2774 df-sbc 2999 df-csb 3094 df-dif 3168 df-un 3170 df-in 3172 df-ss 3179 df-if 3572 df-pw 3618 df-sn 3639 df-pr 3640 df-op 3642 df-uni 3851 df-iun 3929 df-br 4045 df-opab 4106 df-mpt 4107 df-id 4340 df-po 4343 df-iso 4344 df-xp 4681 df-rel 4682 df-cnv 4683 df-co 4684 df-dm 4685 df-rn 4686 df-res 4687 df-ima 4688 df-iota 5232 df-fun 5273 df-fn 5274 df-f 5275 df-fv 5279 df-riota 5899 df-ov 5947 df-oprab 5948 df-mpo 5949 df-1st 6226 df-2nd 6227 df-map 6737 df-pnf 8109 df-mnf 8110 df-xr 8111 df-ltxr 8112 df-le 8113 df-sub 8245 df-neg 8246 df-2 9095 df-xneg 9894 df-xadd 9895 df-psmet 14305 df-xmet 14306 df-bl 14308

This theorem is referenced by: blhalf 14880 blss 14900

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