blss2 - Intuitionistic Logic Explorer

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Theorem blss2 14964

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 8152	. 2 $/\$ $/\$ $/\$ $/\$ $/\$
6		simpr2 1007	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$
7	6	rexrd 8152	. 2 $/\$ $/\$ $/\$ $/\$ $/\$
8	6, 4	resubcld 8483	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$
9		simpr3 1008	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$
10		xmetlecl 14924	. . 3 $/\$ $/\$ $/\$ $/\$
11	1, 2, 3, 8, 9, 10	syl122anc 1259	. 2 $/\$ $/\$ $/\$ $/\$ $/\$
12		rexsub 10005	. . . 4 $/\$
13	6, 4, 12	syl2anc 411	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$
14	9, 13	breqtrrd 4082	. 2 $/\$ $/\$ $/\$ $/\$ $/\$
15	1, 2, 3, 5, 7, 11, 14	xblss2 14962	1 $/\$ $/\$ $/\$ $/\$ $/\$

Colors of variables: wff set class

Syntax hints:

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

wceq 1373

wcel 2177

wss 3170 class class class wbr 4054

cfv 5285 (class class class)co 5962

cr 7954

cle 8138

cmin 8273

cxne 9921

cxad 9922

cxmet 14383

cbl 14385

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 4173 ax-pow 4229 ax-pr 4264 ax-un 4493 ax-setind 4598 ax-cnex 8046 ax-resscn 8047 ax-1cn 8048 ax-1re 8049 ax-icn 8050 ax-addcl 8051 ax-addrcl 8052 ax-mulcl 8053 ax-mulrcl 8054 ax-addcom 8055 ax-mulcom 8056 ax-addass 8057 ax-mulass 8058 ax-distr 8059 ax-i2m1 8060 ax-0lt1 8061 ax-1rid 8062 ax-0id 8063 ax-rnegex 8064 ax-precex 8065 ax-cnre 8066 ax-pre-ltirr 8067 ax-pre-ltwlin 8068 ax-pre-lttrn 8069 ax-pre-apti 8070 ax-pre-ltadd 8071 ax-pre-mulgt0 8072

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 3860 df-iun 3938 df-br 4055 df-opab 4117 df-mpt 4118 df-id 4353 df-po 4356 df-iso 4357 df-xp 4694 df-rel 4695 df-cnv 4696 df-co 4697 df-dm 4698 df-rn 4699 df-res 4700 df-ima 4701 df-iota 5246 df-fun 5287 df-fn 5288 df-f 5289 df-fv 5293 df-riota 5917 df-ov 5965 df-oprab 5966 df-mpo 5967 df-1st 6244 df-2nd 6245 df-map 6755 df-pnf 8139 df-mnf 8140 df-xr 8141 df-ltxr 8142 df-le 8143 df-sub 8275 df-neg 8276 df-2 9125 df-xneg 9924 df-xadd 9925 df-psmet 14390 df-xmet 14391 df-bl 14393

This theorem is referenced by: blhalf 14965 blss 14985

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