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Mirrors > Home > ILE Home > Th. List > aptipr | Unicode version |
Description: Apartness of positive reals is tight. (Contributed by Jim Kingdon, 28-Jan-2020.) |
Ref | Expression |
---|---|
aptipr |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp1 944 |
. . . 4
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2 | simp2 945 |
. . . 4
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3 | ioran 705 |
. . . . . . 7
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4 | 3 | biimpi 119 |
. . . . . 6
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5 | 4 | 3ad2ant3 967 |
. . . . 5
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6 | 5 | simprd 113 |
. . . 4
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7 | aptiprleml 7259 |
. . . 4
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8 | 1, 2, 6, 7 | syl3anc 1175 |
. . 3
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9 | 5 | simpld 111 |
. . . 4
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10 | aptiprleml 7259 |
. . . 4
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11 | 2, 1, 9, 10 | syl3anc 1175 |
. . 3
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12 | 8, 11 | eqssd 3043 |
. 2
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13 | aptiprlemu 7260 |
. . . 4
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14 | 2, 1, 9, 13 | syl3anc 1175 |
. . 3
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15 | aptiprlemu 7260 |
. . . 4
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16 | 1, 2, 6, 15 | syl3anc 1175 |
. . 3
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17 | 14, 16 | eqssd 3043 |
. 2
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18 | preqlu 7092 |
. . 3
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19 | 18 | 3adant3 964 |
. 2
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20 | 12, 17, 19 | mpbir2and 891 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 580 ax-in2 581 ax-io 666 ax-5 1382 ax-7 1383 ax-gen 1384 ax-ie1 1428 ax-ie2 1429 ax-8 1441 ax-10 1442 ax-11 1443 ax-i12 1444 ax-bndl 1445 ax-4 1446 ax-13 1450 ax-14 1451 ax-17 1465 ax-i9 1469 ax-ial 1473 ax-i5r 1474 ax-ext 2071 ax-coll 3960 ax-sep 3963 ax-nul 3971 ax-pow 4015 ax-pr 4045 ax-un 4269 ax-setind 4366 ax-iinf 4416 |
This theorem depends on definitions: df-bi 116 df-dc 782 df-3or 926 df-3an 927 df-tru 1293 df-fal 1296 df-nf 1396 df-sb 1694 df-eu 1952 df-mo 1953 df-clab 2076 df-cleq 2082 df-clel 2085 df-nfc 2218 df-ne 2257 df-ral 2365 df-rex 2366 df-reu 2367 df-rab 2369 df-v 2622 df-sbc 2842 df-csb 2935 df-dif 3002 df-un 3004 df-in 3006 df-ss 3013 df-nul 3288 df-pw 3435 df-sn 3456 df-pr 3457 df-op 3459 df-uni 3660 df-int 3695 df-iun 3738 df-br 3852 df-opab 3906 df-mpt 3907 df-tr 3943 df-eprel 4125 df-id 4129 df-po 4132 df-iso 4133 df-iord 4202 df-on 4204 df-suc 4207 df-iom 4419 df-xp 4458 df-rel 4459 df-cnv 4460 df-co 4461 df-dm 4462 df-rn 4463 df-res 4464 df-ima 4465 df-iota 4993 df-fun 5030 df-fn 5031 df-f 5032 df-f1 5033 df-fo 5034 df-f1o 5035 df-fv 5036 df-ov 5669 df-oprab 5670 df-mpt2 5671 df-1st 5925 df-2nd 5926 df-recs 6084 df-irdg 6149 df-1o 6195 df-2o 6196 df-oadd 6199 df-omul 6200 df-er 6306 df-ec 6308 df-qs 6312 df-ni 6924 df-pli 6925 df-mi 6926 df-lti 6927 df-plpq 6964 df-mpq 6965 df-enq 6967 df-nqqs 6968 df-plqqs 6969 df-mqqs 6970 df-1nqqs 6971 df-rq 6972 df-ltnqqs 6973 df-enq0 7044 df-nq0 7045 df-0nq0 7046 df-plq0 7047 df-mq0 7048 df-inp 7086 df-iltp 7090 |
This theorem is referenced by: aptisr 7385 |
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