blss2ps - Intuitionistic Logic Explorer

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Theorem blss2ps 14642

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 8076	. 2 PsMet $/\$ $/\$ $/\$ $/\$ $/\$
6		simpr2 1006	. . 3 PsMet $/\$ $/\$ $/\$ $/\$ $/\$
7	6	rexrd 8076	. 2 PsMet $/\$ $/\$ $/\$ $/\$ $/\$
8	6, 4	resubcld 8407	. . 3 PsMet $/\$ $/\$ $/\$ $/\$ $/\$
9		simpr3 1007	. . 3 PsMet $/\$ $/\$ $/\$ $/\$ $/\$
10		psmetlecl 14570	. . 3 PsMet $/\$ $/\$ $/\$ $/\$
11	1, 2, 3, 8, 9, 10	syl122anc 1258	. 2 PsMet $/\$ $/\$ $/\$ $/\$ $/\$
12		rexsub 9928	. . . 4 $/\$
13	6, 4, 12	syl2anc 411	. . 3 PsMet $/\$ $/\$ $/\$ $/\$ $/\$
14	9, 13	breqtrrd 4061	. 2 PsMet $/\$ $/\$ $/\$ $/\$ $/\$
15	1, 2, 3, 5, 7, 11, 14	xblss2ps 14640	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 4033

cfv 5258 (class class class)co 5922

cr 7878

cle 8062

cmin 8197

cxne 9844

cxad 9845 PsMetcpsmet 14091

cbl 14094

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 4151 ax-pow 4207 ax-pr 4242 ax-un 4468 ax-setind 4573 ax-cnex 7970 ax-resscn 7971 ax-1cn 7972 ax-1re 7973 ax-icn 7974 ax-addcl 7975 ax-addrcl 7976 ax-mulcl 7977 ax-mulrcl 7978 ax-addcom 7979 ax-mulcom 7980 ax-addass 7981 ax-mulass 7982 ax-distr 7983 ax-i2m1 7984 ax-0lt1 7985 ax-1rid 7986 ax-0id 7987 ax-rnegex 7988 ax-precex 7989 ax-cnre 7990 ax-pre-ltirr 7991 ax-pre-ltwlin 7992 ax-pre-lttrn 7993 ax-pre-apti 7994 ax-pre-ltadd 7995 ax-pre-mulgt0 7996

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 3562 df-pw 3607 df-sn 3628 df-pr 3629 df-op 3631 df-uni 3840 df-iun 3918 df-br 4034 df-opab 4095 df-mpt 4096 df-id 4328 df-po 4331 df-iso 4332 df-xp 4669 df-rel 4670 df-cnv 4671 df-co 4672 df-dm 4673 df-rn 4674 df-res 4675 df-ima 4676 df-iota 5219 df-fun 5260 df-fn 5261 df-f 5262 df-fv 5266 df-riota 5877 df-ov 5925 df-oprab 5926 df-mpo 5927 df-1st 6198 df-2nd 6199 df-map 6709 df-pnf 8063 df-mnf 8064 df-xr 8065 df-ltxr 8066 df-le 8067 df-sub 8199 df-neg 8200 df-2 9049 df-xneg 9847 df-xadd 9848 df-psmet 14099 df-bl 14102

This theorem is referenced by: blssps 14663

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