blss2ps - Intuitionistic Logic Explorer

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Theorem blss2ps 12614

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 985	. 2 PsMet $/\$ $/\$ $/\$ $/\$ $/\$ PsMet
2		simpl2 986	. 2 PsMet $/\$ $/\$ $/\$ $/\$ $/\$
3		simpl3 987	. 2 PsMet $/\$ $/\$ $/\$ $/\$ $/\$
4		simpr1 988	. . 3 PsMet $/\$ $/\$ $/\$ $/\$ $/\$
5	4	rexrd 7839	. 2 PsMet $/\$ $/\$ $/\$ $/\$ $/\$
6		simpr2 989	. . 3 PsMet $/\$ $/\$ $/\$ $/\$ $/\$
7	6	rexrd 7839	. 2 PsMet $/\$ $/\$ $/\$ $/\$ $/\$
8	6, 4	resubcld 8167	. . 3 PsMet $/\$ $/\$ $/\$ $/\$ $/\$
9		simpr3 990	. . 3 PsMet $/\$ $/\$ $/\$ $/\$ $/\$
10		psmetlecl 12542	. . 3 PsMet $/\$ $/\$ $/\$ $/\$
11	1, 2, 3, 8, 9, 10	syl122anc 1226	. 2 PsMet $/\$ $/\$ $/\$ $/\$ $/\$
12		rexsub 9666	. . . 4 $/\$
13	6, 4, 12	syl2anc 409	. . 3 PsMet $/\$ $/\$ $/\$ $/\$ $/\$
14	9, 13	breqtrrd 3964	. 2 PsMet $/\$ $/\$ $/\$ $/\$ $/\$
15	1, 2, 3, 5, 7, 11, 14	xblss2ps 12612	1 PsMet $/\$ $/\$ $/\$ $/\$ $/\$

Colors of variables: wff set class

Syntax hints:

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

wceq 1332

wcel 1481

wss 3076 class class class wbr 3937

cfv 5131 (class class class)co 5782

cr 7643

cle 7825

cmin 7957

cxne 9586

cxad 9587 PsMetcpsmet 12187

cbl 12190

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 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-13 1492 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-sep 4054 ax-pow 4106 ax-pr 4139 ax-un 4363 ax-setind 4460 ax-cnex 7735 ax-resscn 7736 ax-1cn 7737 ax-1re 7738 ax-icn 7739 ax-addcl 7740 ax-addrcl 7741 ax-mulcl 7742 ax-mulrcl 7743 ax-addcom 7744 ax-mulcom 7745 ax-addass 7746 ax-mulass 7747 ax-distr 7748 ax-i2m1 7749 ax-0lt1 7750 ax-1rid 7751 ax-0id 7752 ax-rnegex 7753 ax-precex 7754 ax-cnre 7755 ax-pre-ltirr 7756 ax-pre-ltwlin 7757 ax-pre-lttrn 7758 ax-pre-apti 7759 ax-pre-ltadd 7760 ax-pre-mulgt0 7761

This theorem depends on definitions: df-bi 116 df-stab 817 df-dc 821 df-3or 964 df-3an 965 df-tru 1335 df-fal 1338 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ne 2310 df-nel 2405 df-ral 2422 df-rex 2423 df-reu 2424 df-rab 2426 df-v 2691 df-sbc 2914 df-csb 3008 df-dif 3078 df-un 3080 df-in 3082 df-ss 3089 df-if 3480 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-iun 3823 df-br 3938 df-opab 3998 df-mpt 3999 df-id 4223 df-po 4226 df-iso 4227 df-xp 4553 df-rel 4554 df-cnv 4555 df-co 4556 df-dm 4557 df-rn 4558 df-res 4559 df-ima 4560 df-iota 5096 df-fun 5133 df-fn 5134 df-f 5135 df-fv 5139 df-riota 5738 df-ov 5785 df-oprab 5786 df-mpo 5787 df-1st 6046 df-2nd 6047 df-map 6552 df-pnf 7826 df-mnf 7827 df-xr 7828 df-ltxr 7829 df-le 7830 df-sub 7959 df-neg 7960 df-2 8803 df-xneg 9589 df-xadd 9590 df-psmet 12195 df-bl 12198

This theorem is referenced by: blssps 12635

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