blss2ps - Intuitionistic Logic Explorer

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Theorem blss2ps 14953

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 1003	. 2 PsMet $/\$ $/\$ $/\$ $/\$ $/\$ PsMet
2		simpl2 1004	. 2 PsMet $/\$ $/\$ $/\$ $/\$ $/\$
3		simpl3 1005	. 2 PsMet $/\$ $/\$ $/\$ $/\$ $/\$
4		simpr1 1006	. . 3 PsMet $/\$ $/\$ $/\$ $/\$ $/\$
5	4	rexrd 8142	. 2 PsMet $/\$ $/\$ $/\$ $/\$ $/\$
6		simpr2 1007	. . 3 PsMet $/\$ $/\$ $/\$ $/\$ $/\$
7	6	rexrd 8142	. 2 PsMet $/\$ $/\$ $/\$ $/\$ $/\$
8	6, 4	resubcld 8473	. . 3 PsMet $/\$ $/\$ $/\$ $/\$ $/\$
9		simpr3 1008	. . 3 PsMet $/\$ $/\$ $/\$ $/\$ $/\$
10		psmetlecl 14881	. . 3 PsMet $/\$ $/\$ $/\$ $/\$
11	1, 2, 3, 8, 9, 10	syl122anc 1259	. 2 PsMet $/\$ $/\$ $/\$ $/\$ $/\$
12		rexsub 9995	. . . 4 $/\$
13	6, 4, 12	syl2anc 411	. . 3 PsMet $/\$ $/\$ $/\$ $/\$ $/\$
14	9, 13	breqtrrd 4079	. 2 PsMet $/\$ $/\$ $/\$ $/\$ $/\$
15	1, 2, 3, 5, 7, 11, 14	xblss2ps 14951	1 PsMet $/\$ $/\$ $/\$ $/\$ $/\$

Colors of variables: wff set class

Syntax hints:

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

wceq 1373

wcel 2177

wss 3170 class class class wbr 4051

cfv 5280 (class class class)co 5957

cr 7944

cle 8128

cmin 8263

cxne 9911

cxad 9912 PsMetcpsmet 14372

cbl 14375

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 1471 ax-7 1472 ax-gen 1473 ax-ie1 1517 ax-ie2 1518 ax-8 1528 ax-10 1529 ax-11 1530 ax-i12 1531 ax-bndl 1533 ax-4 1534 ax-17 1550 ax-i9 1554 ax-ial 1558 ax-i5r 1559 ax-13 2179 ax-14 2180 ax-ext 2188 ax-sep 4170 ax-pow 4226 ax-pr 4261 ax-un 4488 ax-setind 4593 ax-cnex 8036 ax-resscn 8037 ax-1cn 8038 ax-1re 8039 ax-icn 8040 ax-addcl 8041 ax-addrcl 8042 ax-mulcl 8043 ax-mulrcl 8044 ax-addcom 8045 ax-mulcom 8046 ax-addass 8047 ax-mulass 8048 ax-distr 8049 ax-i2m1 8050 ax-0lt1 8051 ax-1rid 8052 ax-0id 8053 ax-rnegex 8054 ax-precex 8055 ax-cnre 8056 ax-pre-ltirr 8057 ax-pre-ltwlin 8058 ax-pre-lttrn 8059 ax-pre-apti 8060 ax-pre-ltadd 8061 ax-pre-mulgt0 8062

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 1485 df-sb 1787 df-eu 2058 df-mo 2059 df-clab 2193 df-cleq 2199 df-clel 2202 df-nfc 2338 df-ne 2378 df-nel 2473 df-ral 2490 df-rex 2491 df-reu 2492 df-rab 2494 df-v 2775 df-sbc 3003 df-csb 3098 df-dif 3172 df-un 3174 df-in 3176 df-ss 3183 df-if 3576 df-pw 3623 df-sn 3644 df-pr 3645 df-op 3647 df-uni 3857 df-iun 3935 df-br 4052 df-opab 4114 df-mpt 4115 df-id 4348 df-po 4351 df-iso 4352 df-xp 4689 df-rel 4690 df-cnv 4691 df-co 4692 df-dm 4693 df-rn 4694 df-res 4695 df-ima 4696 df-iota 5241 df-fun 5282 df-fn 5283 df-f 5284 df-fv 5288 df-riota 5912 df-ov 5960 df-oprab 5961 df-mpo 5962 df-1st 6239 df-2nd 6240 df-map 6750 df-pnf 8129 df-mnf 8130 df-xr 8131 df-ltxr 8132 df-le 8133 df-sub 8265 df-neg 8266 df-2 9115 df-xneg 9914 df-xadd 9915 df-psmet 14380 df-bl 14383

This theorem is referenced by: blssps 14974

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