blss2 - Intuitionistic Logic Explorer

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Theorem blss2 12778

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 985	. 2 $/\$ $/\$ $/\$ $/\$ $/\$
2		simpl2 986	. 2 $/\$ $/\$ $/\$ $/\$ $/\$
3		simpl3 987	. 2 $/\$ $/\$ $/\$ $/\$ $/\$
4		simpr1 988	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$
5	4	rexrd 7921	. 2 $/\$ $/\$ $/\$ $/\$ $/\$
6		simpr2 989	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$
7	6	rexrd 7921	. 2 $/\$ $/\$ $/\$ $/\$ $/\$
8	6, 4	resubcld 8250	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$
9		simpr3 990	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$
10		xmetlecl 12738	. . 3 $/\$ $/\$ $/\$ $/\$
11	1, 2, 3, 8, 9, 10	syl122anc 1229	. 2 $/\$ $/\$ $/\$ $/\$ $/\$
12		rexsub 9750	. . . 4 $/\$
13	6, 4, 12	syl2anc 409	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$
14	9, 13	breqtrrd 3992	. 2 $/\$ $/\$ $/\$ $/\$ $/\$
15	1, 2, 3, 5, 7, 11, 14	xblss2 12776	1 $/\$ $/\$ $/\$ $/\$ $/\$

Colors of variables: wff set class

Syntax hints:

wi 4 $/\$ wa 103 $/\$ w3a 963

wceq 1335

wcel 2128

wss 3102 class class class wbr 3965

cfv 5169 (class class class)co 5821

cr 7725

cle 7907

cmin 8040

cxne 9669

cxad 9670

cxmet 12351

cbl 12353

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-13 2130 ax-14 2131 ax-ext 2139 ax-sep 4082 ax-pow 4135 ax-pr 4169 ax-un 4393 ax-setind 4495 ax-cnex 7817 ax-resscn 7818 ax-1cn 7819 ax-1re 7820 ax-icn 7821 ax-addcl 7822 ax-addrcl 7823 ax-mulcl 7824 ax-mulrcl 7825 ax-addcom 7826 ax-mulcom 7827 ax-addass 7828 ax-mulass 7829 ax-distr 7830 ax-i2m1 7831 ax-0lt1 7832 ax-1rid 7833 ax-0id 7834 ax-rnegex 7835 ax-precex 7836 ax-cnre 7837 ax-pre-ltirr 7838 ax-pre-ltwlin 7839 ax-pre-lttrn 7840 ax-pre-apti 7841 ax-pre-ltadd 7842 ax-pre-mulgt0 7843

This theorem depends on definitions: df-bi 116 df-stab 817 df-dc 821 df-3or 964 df-3an 965 df-tru 1338 df-fal 1341 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ne 2328 df-nel 2423 df-ral 2440 df-rex 2441 df-reu 2442 df-rab 2444 df-v 2714 df-sbc 2938 df-csb 3032 df-dif 3104 df-un 3106 df-in 3108 df-ss 3115 df-if 3506 df-pw 3545 df-sn 3566 df-pr 3567 df-op 3569 df-uni 3773 df-iun 3851 df-br 3966 df-opab 4026 df-mpt 4027 df-id 4253 df-po 4256 df-iso 4257 df-xp 4591 df-rel 4592 df-cnv 4593 df-co 4594 df-dm 4595 df-rn 4596 df-res 4597 df-ima 4598 df-iota 5134 df-fun 5171 df-fn 5172 df-f 5173 df-fv 5177 df-riota 5777 df-ov 5824 df-oprab 5825 df-mpo 5826 df-1st 6085 df-2nd 6086 df-map 6592 df-pnf 7908 df-mnf 7909 df-xr 7910 df-ltxr 7911 df-le 7912 df-sub 8042 df-neg 8043 df-2 8886 df-xneg 9672 df-xadd 9673 df-psmet 12358 df-xmet 12359 df-bl 12361

This theorem is referenced by: blhalf 12779 blss 12799

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