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Mirrors > Home > ILE Home > Th. List > prnminu | Unicode version |
Description: An upper cut has no smallest member. (Contributed by Jim Kingdon, 7-Nov-2019.) |
Ref | Expression |
---|---|
prnminu |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elprnqu 7495 |
. . . . 5
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2 | elinp 7487 |
. . . . . . . 8
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3 | simpr1r 1056 |
. . . . . . . 8
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4 | 2, 3 | sylbi 121 |
. . . . . . 7
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5 | eleq1 2250 |
. . . . . . . . 9
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6 | breq2 4019 |
. . . . . . . . . . 11
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7 | 6 | anbi1d 465 |
. . . . . . . . . 10
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8 | 7 | rexbidv 2488 |
. . . . . . . . 9
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9 | 5, 8 | bibi12d 235 |
. . . . . . . 8
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10 | 9 | rspcv 2849 |
. . . . . . 7
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11 | biimp 118 |
. . . . . . 7
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12 | 4, 10, 11 | syl56 34 |
. . . . . 6
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13 | 12 | impd 254 |
. . . . 5
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14 | 1, 13 | mpcom 36 |
. . . 4
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15 | df-rex 2471 |
. . . 4
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16 | 14, 15 | sylib 122 |
. . 3
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17 | ltrelnq 7378 |
. . . . . . . . 9
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18 | 17 | brel 4690 |
. . . . . . . 8
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19 | 18 | simpld 112 |
. . . . . . 7
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20 | 19 | pm4.71ri 392 |
. . . . . 6
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21 | 20 | anbi1i 458 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
22 | ancom 266 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
23 | anass 401 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
24 | 21, 22, 23 | 3bitr3i 210 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
25 | 24 | exbii 1615 |
. . 3
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26 | 16, 25 | sylibr 134 |
. 2
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27 | df-rex 2471 |
. 2
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
28 | 26, 27 | sylibr 134 |
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 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1457 ax-7 1458 ax-gen 1459 ax-ie1 1503 ax-ie2 1504 ax-8 1514 ax-10 1515 ax-11 1516 ax-i12 1517 ax-bndl 1519 ax-4 1520 ax-17 1536 ax-i9 1540 ax-ial 1544 ax-i5r 1545 ax-13 2160 ax-14 2161 ax-ext 2169 ax-coll 4130 ax-sep 4133 ax-pow 4186 ax-pr 4221 ax-un 4445 ax-iinf 4599 |
This theorem depends on definitions: df-bi 117 df-3an 981 df-tru 1366 df-nf 1471 df-sb 1773 df-eu 2039 df-mo 2040 df-clab 2174 df-cleq 2180 df-clel 2183 df-nfc 2318 df-ral 2470 df-rex 2471 df-reu 2472 df-rab 2474 df-v 2751 df-sbc 2975 df-csb 3070 df-dif 3143 df-un 3145 df-in 3147 df-ss 3154 df-pw 3589 df-sn 3610 df-pr 3611 df-op 3613 df-uni 3822 df-int 3857 df-iun 3900 df-br 4016 df-opab 4077 df-mpt 4078 df-id 4305 df-iom 4602 df-xp 4644 df-rel 4645 df-cnv 4646 df-co 4647 df-dm 4648 df-rn 4649 df-res 4650 df-ima 4651 df-iota 5190 df-fun 5230 df-fn 5231 df-f 5232 df-f1 5233 df-fo 5234 df-f1o 5235 df-fv 5236 df-qs 6555 df-ni 7317 df-nqqs 7361 df-ltnqqs 7366 df-inp 7479 |
This theorem is referenced by: genprndu 7535 nqpru 7565 1idpru 7604 ltsopr 7609 ltexprlemopu 7616 ltexprlemru 7625 addcanprlemu 7628 recexprlemloc 7644 recexprlem1ssu 7647 aptiprlemu 7653 |
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