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Mirrors > Home > ILE Home > Th. List > ordpwsucexmid | Unicode version |
Description: The subset in ordpwsucss 4490 cannot be equality. That is, strengthening it to equality implies excluded middle. (Contributed by Jim Kingdon, 30-Jul-2019.) |
Ref | Expression |
---|---|
ordpwsucexmid.1 |
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Ref | Expression |
---|---|
ordpwsucexmid |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0elpw 4096 |
. . . . 5
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2 | 0elon 4322 |
. . . . 5
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3 | elin 3264 |
. . . . 5
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4 | 1, 2, 3 | mpbir2an 927 |
. . . 4
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5 | ordtriexmidlem 4443 |
. . . . 5
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6 | suceq 4332 |
. . . . . . 7
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7 | pweq 3518 |
. . . . . . . 8
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8 | 7 | ineq1d 3281 |
. . . . . . 7
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9 | 6, 8 | eqeq12d 2155 |
. . . . . 6
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10 | ordpwsucexmid.1 |
. . . . . 6
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11 | 9, 10 | vtoclri 2764 |
. . . . 5
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12 | 5, 11 | ax-mp 5 |
. . . 4
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13 | 4, 12 | eleqtrri 2216 |
. . 3
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14 | elsuci 4333 |
. . 3
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15 | 13, 14 | ax-mp 5 |
. 2
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16 | 0ex 4063 |
. . . . . 6
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17 | 16 | snid 3563 |
. . . . 5
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18 | biidd 171 |
. . . . . 6
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19 | 18 | elrab3 2845 |
. . . . 5
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20 | 17, 19 | ax-mp 5 |
. . . 4
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21 | 20 | biimpi 119 |
. . 3
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22 | ordtriexmidlem2 4444 |
. . . 4
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23 | 22 | eqcoms 2143 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
24 | 21, 23 | orim12i 749 |
. 2
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25 | 15, 24 | ax-mp 5 |
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-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-sep 4054 ax-nul 4062 ax-pow 4106 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-nf 1438 df-sb 1737 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ral 2422 df-rex 2423 df-rab 2426 df-v 2691 df-dif 3078 df-un 3080 df-in 3082 df-ss 3089 df-nul 3369 df-pw 3517 df-sn 3538 df-uni 3745 df-tr 4035 df-iord 4296 df-on 4298 df-suc 4301 |
This theorem is referenced by: (None) |
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