blss2ps - Intuitionistic Logic Explorer

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Theorem blss2ps 14726

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.) (Revised by Thierry Arnoux, 11-Mar-2018.)

**Assertion**
Ref	Expression
blss2ps	PsMet $/\$ $/\$ $/\$ $/\$ $/\$

**Proof of Theorem blss2ps**
Step	Hyp	Ref	Expression
1		simpl1 1002	. 2 PsMet $/\$ $/\$ $/\$ $/\$ $/\$ PsMet
2		simpl2 1003	. 2 PsMet $/\$ $/\$ $/\$ $/\$ $/\$
3		simpl3 1004	. 2 PsMet $/\$ $/\$ $/\$ $/\$ $/\$
4		simpr1 1005	. . 3 PsMet $/\$ $/\$ $/\$ $/\$ $/\$
5	4	rexrd 8093	. 2 PsMet $/\$ $/\$ $/\$ $/\$ $/\$
6		simpr2 1006	. . 3 PsMet $/\$ $/\$ $/\$ $/\$ $/\$
7	6	rexrd 8093	. 2 PsMet $/\$ $/\$ $/\$ $/\$ $/\$
8	6, 4	resubcld 8424	. . 3 PsMet $/\$ $/\$ $/\$ $/\$ $/\$
9		simpr3 1007	. . 3 PsMet $/\$ $/\$ $/\$ $/\$ $/\$
10		psmetlecl 14654	. . 3 PsMet $/\$ $/\$ $/\$ $/\$
11	1, 2, 3, 8, 9, 10	syl122anc 1258	. 2 PsMet $/\$ $/\$ $/\$ $/\$ $/\$
12		rexsub 9945	. . . 4 $/\$
13	6, 4, 12	syl2anc 411	. . 3 PsMet $/\$ $/\$ $/\$ $/\$ $/\$
14	9, 13	breqtrrd 4062	. 2 PsMet $/\$ $/\$ $/\$ $/\$ $/\$
15	1, 2, 3, 5, 7, 11, 14	xblss2ps 14724	1 PsMet $/\$ $/\$ $/\$ $/\$ $/\$

Colors of variables: wff set class

Syntax hints:

wi 4 $/\$ wa 104 $/\$ w3a 980

wceq 1364

wcel 2167

wss 3157 class class class wbr 4034

cfv 5259 (class class class)co 5925

cr 7895

cle 8079

cmin 8214

cxne 9861

cxad 9862 PsMetcpsmet 14167

cbl 14170

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 710 ax-5 1461 ax-7 1462 ax-gen 1463 ax-ie1 1507 ax-ie2 1508 ax-8 1518 ax-10 1519 ax-11 1520 ax-i12 1521 ax-bndl 1523 ax-4 1524 ax-17 1540 ax-i9 1544 ax-ial 1548 ax-i5r 1549 ax-13 2169 ax-14 2170 ax-ext 2178 ax-sep 4152 ax-pow 4208 ax-pr 4243 ax-un 4469 ax-setind 4574 ax-cnex 7987 ax-resscn 7988 ax-1cn 7989 ax-1re 7990 ax-icn 7991 ax-addcl 7992 ax-addrcl 7993 ax-mulcl 7994 ax-mulrcl 7995 ax-addcom 7996 ax-mulcom 7997 ax-addass 7998 ax-mulass 7999 ax-distr 8000 ax-i2m1 8001 ax-0lt1 8002 ax-1rid 8003 ax-0id 8004 ax-rnegex 8005 ax-precex 8006 ax-cnre 8007 ax-pre-ltirr 8008 ax-pre-ltwlin 8009 ax-pre-lttrn 8010 ax-pre-apti 8011 ax-pre-ltadd 8012 ax-pre-mulgt0 8013

This theorem depends on definitions: df-bi 117 df-stab 832 df-dc 836 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1475 df-sb 1777 df-eu 2048 df-mo 2049 df-clab 2183 df-cleq 2189 df-clel 2192 df-nfc 2328 df-ne 2368 df-nel 2463 df-ral 2480 df-rex 2481 df-reu 2482 df-rab 2484 df-v 2765 df-sbc 2990 df-csb 3085 df-dif 3159 df-un 3161 df-in 3163 df-ss 3170 df-if 3563 df-pw 3608 df-sn 3629 df-pr 3630 df-op 3632 df-uni 3841 df-iun 3919 df-br 4035 df-opab 4096 df-mpt 4097 df-id 4329 df-po 4332 df-iso 4333 df-xp 4670 df-rel 4671 df-cnv 4672 df-co 4673 df-dm 4674 df-rn 4675 df-res 4676 df-ima 4677 df-iota 5220 df-fun 5261 df-fn 5262 df-f 5263 df-fv 5267 df-riota 5880 df-ov 5928 df-oprab 5929 df-mpo 5930 df-1st 6207 df-2nd 6208 df-map 6718 df-pnf 8080 df-mnf 8081 df-xr 8082 df-ltxr 8083 df-le 8084 df-sub 8216 df-neg 8217 df-2 9066 df-xneg 9864 df-xadd 9865 df-psmet 14175 df-bl 14178

This theorem is referenced by: blssps 14747

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