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Mirrors > Home > MPE Home > Th. List > ssbl | Unicode version |
Description: The size of a ball increases monotonically with its radius. (Contributed by NM, 20-Sep-2007.) (Revised by Mario Carneiro, 24-Aug-2015.) |
Ref | Expression |
---|---|
ssbl |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp1l 981 |
. 2
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2 | simp1r 982 |
. 2
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3 | simp2l 983 |
. 2
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4 | simp2r 984 |
. 2
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5 | xmet0 18329 |
. . . 4
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6 | 5 | 3ad2ant1 978 |
. . 3
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7 | 0re 9051 |
. . 3
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8 | 6, 7 | syl6eqel 2496 |
. 2
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9 | simp3 959 |
. . . 4
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10 | xsubge0 10800 |
. . . . 5
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11 | 4, 3, 10 | syl2anc 643 |
. . . 4
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12 | 9, 11 | mpbird 224 |
. . 3
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13 | 6, 12 | eqbrtrd 4196 |
. 2
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14 | 1, 2, 2, 3, 4, 8, 13 | xblss2 18389 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem is referenced by: blss 18412 ssblex 18415 blssec 18422 metequiv2 18497 met1stc 18508 met2ndci 18509 metdstri 18838 xlebnum 18947 iscmet3lem2 19202 caubl 19217 heiborlem8 26421 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-3 7 ax-mp 8 ax-gen 1552 ax-5 1563 ax-17 1623 ax-9 1662 ax-8 1683 ax-13 1723 ax-14 1725 ax-6 1740 ax-7 1745 ax-11 1757 ax-12 1946 ax-ext 2389 ax-sep 4294 ax-nul 4302 ax-pow 4341 ax-pr 4367 ax-un 4664 ax-cnex 9006 ax-resscn 9007 ax-1cn 9008 ax-icn 9009 ax-addcl 9010 ax-addrcl 9011 ax-mulcl 9012 ax-mulrcl 9013 ax-mulcom 9014 ax-addass 9015 ax-mulass 9016 ax-distr 9017 ax-i2m1 9018 ax-1ne0 9019 ax-1rid 9020 ax-rnegex 9021 ax-rrecex 9022 ax-cnre 9023 ax-pre-lttri 9024 ax-pre-lttrn 9025 ax-pre-ltadd 9026 ax-pre-mulgt0 9027 |
This theorem depends on definitions: df-bi 178 df-or 360 df-an 361 df-3or 937 df-3an 938 df-tru 1325 df-ex 1548 df-nf 1551 df-sb 1656 df-eu 2262 df-mo 2263 df-clab 2395 df-cleq 2401 df-clel 2404 df-nfc 2533 df-ne 2573 df-nel 2574 df-ral 2675 df-rex 2676 df-reu 2677 df-rmo 2678 df-rab 2679 df-v 2922 df-sbc 3126 df-csb 3216 df-dif 3287 df-un 3289 df-in 3291 df-ss 3298 df-nul 3593 df-if 3704 df-pw 3765 df-sn 3784 df-pr 3785 df-op 3787 df-uni 3980 df-iun 4059 df-br 4177 df-opab 4231 df-mpt 4232 df-id 4462 df-po 4467 df-so 4468 df-xp 4847 df-rel 4848 df-cnv 4849 df-co 4850 df-dm 4851 df-rn 4852 df-res 4853 df-ima 4854 df-iota 5381 df-fun 5419 df-fn 5420 df-f 5421 df-f1 5422 df-fo 5423 df-f1o 5424 df-fv 5425 df-ov 6047 df-oprab 6048 df-mpt2 6049 df-1st 6312 df-2nd 6313 df-riota 6512 df-er 6868 df-map 6983 df-en 7073 df-dom 7074 df-sdom 7075 df-pnf 9082 df-mnf 9083 df-xr 9084 df-ltxr 9085 df-le 9086 df-sub 9253 df-neg 9254 df-div 9638 df-2 10018 df-rp 10573 df-xneg 10670 df-xadd 10671 df-xmul 10672 df-psmet 16653 df-xmet 16654 df-bl 16656 |
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