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Mirrors > Home > ILE Home > Th. List > genpcuu | Unicode version |
Description: Upward closure of an operation on positive reals. (Contributed by Jim Kingdon, 8-Nov-2019.) |
Ref | Expression |
---|---|
genpelvl.1 |
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genpelvl.2 |
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genpcuu.2 |
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Ref | Expression |
---|---|
genpcuu |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ltrelnq 7197 |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
2 | 1 | brel 4599 |
. . . . . 6
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3 | 2 | simprd 113 |
. . . . 5
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4 | genpelvl.1 |
. . . . . . . . 9
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5 | genpelvl.2 |
. . . . . . . . 9
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6 | 4, 5 | genpelvu 7345 |
. . . . . . . 8
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7 | 6 | adantr 274 |
. . . . . . 7
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8 | breq1 3940 |
. . . . . . . . . . . . 13
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9 | 8 | biimpd 143 |
. . . . . . . . . . . 12
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10 | genpcuu.2 |
. . . . . . . . . . . 12
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11 | 9, 10 | sylan9r 408 |
. . . . . . . . . . 11
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12 | 11 | exp31 362 |
. . . . . . . . . 10
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13 | 12 | an4s 578 |
. . . . . . . . 9
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14 | 13 | impancom 258 |
. . . . . . . 8
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15 | 14 | rexlimdvv 2559 |
. . . . . . 7
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16 | 7, 15 | sylbid 149 |
. . . . . 6
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17 | 16 | ex 114 |
. . . . 5
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18 | 3, 17 | syl5 32 |
. . . 4
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19 | 18 | com34 83 |
. . 3
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20 | 19 | pm2.43d 50 |
. 2
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21 | 20 | com23 78 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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-coll 4051 ax-sep 4054 ax-pow 4106 ax-pr 4139 ax-un 4363 ax-setind 4460 ax-iinf 4510 |
This theorem depends on definitions: df-bi 116 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-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-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-int 3780 df-iun 3823 df-br 3938 df-opab 3998 df-mpt 3999 df-id 4223 df-iom 4513 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-f1 5136 df-fo 5137 df-f1o 5138 df-fv 5139 df-ov 5785 df-oprab 5786 df-mpo 5787 df-1st 6046 df-2nd 6047 df-qs 6443 df-ni 7136 df-nqqs 7180 df-ltnqqs 7185 df-inp 7298 |
This theorem is referenced by: genprndu 7354 |
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