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Mirrors > Home > ILE Home > Th. List > poirr2 | Unicode version |
Description: A partial order relation is irreflexive. (Contributed by Mario Carneiro, 2-Nov-2015.) |
Ref | Expression |
---|---|
poirr2 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | relres 4855 |
. . . 4
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2 | relin2 4666 |
. . . 4
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3 | 1, 2 | mp1i 10 |
. . 3
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4 | df-br 3938 |
. . . . 5
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5 | brin 3988 |
. . . . 5
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6 | 4, 5 | bitr3i 185 |
. . . 4
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7 | vex 2692 |
. . . . . . . . 9
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8 | 7 | brres 4833 |
. . . . . . . 8
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9 | poirr 4237 |
. . . . . . . . . . 11
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10 | 7 | ideq 4699 |
. . . . . . . . . . . . 13
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11 | breq2 3941 |
. . . . . . . . . . . . 13
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12 | 10, 11 | sylbi 120 |
. . . . . . . . . . . 12
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13 | 12 | notbid 657 |
. . . . . . . . . . 11
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14 | 9, 13 | syl5ibcom 154 |
. . . . . . . . . 10
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15 | 14 | expimpd 361 |
. . . . . . . . 9
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16 | 15 | ancomsd 267 |
. . . . . . . 8
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17 | 8, 16 | syl5bi 151 |
. . . . . . 7
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18 | 17 | con2d 614 |
. . . . . 6
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19 | imnan 680 |
. . . . . 6
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20 | 18, 19 | sylib 121 |
. . . . 5
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21 | 20 | pm2.21d 609 |
. . . 4
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22 | 6, 21 | syl5bi 151 |
. . 3
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23 | 3, 22 | relssdv 4639 |
. 2
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24 | ss0 3408 |
. 2
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25 | 23, 24 | syl 14 |
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-pow 4106 ax-pr 4139 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 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-ral 2422 df-rex 2423 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-pr 3539 df-op 3541 df-br 3938 df-opab 3998 df-id 4223 df-po 4226 df-xp 4553 df-rel 4554 df-res 4559 |
This theorem is referenced by: (None) |
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