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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 4953 |
. . . 4
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2 | relin2 4763 |
. . . 4
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3 | 1, 2 | mp1i 10 |
. . 3
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4 | df-br 4019 |
. . . . 5
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5 | brin 4070 |
. . . . 5
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6 | 4, 5 | bitr3i 186 |
. . . 4
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7 | vex 2755 |
. . . . . . . . 9
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8 | 7 | brres 4931 |
. . . . . . . 8
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9 | poirr 4325 |
. . . . . . . . . . 11
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10 | 7 | ideq 4797 |
. . . . . . . . . . . . 13
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11 | breq2 4022 |
. . . . . . . . . . . . 13
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12 | 10, 11 | sylbi 121 |
. . . . . . . . . . . 12
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13 | 12 | notbid 668 |
. . . . . . . . . . 11
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14 | 9, 13 | syl5ibcom 155 |
. . . . . . . . . 10
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15 | 14 | expimpd 363 |
. . . . . . . . 9
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16 | 15 | ancomsd 269 |
. . . . . . . 8
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17 | 8, 16 | biimtrid 152 |
. . . . . . 7
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18 | 17 | con2d 625 |
. . . . . 6
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19 | imnan 691 |
. . . . . 6
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20 | 18, 19 | sylib 122 |
. . . . 5
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21 | 20 | pm2.21d 620 |
. . . 4
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22 | 6, 21 | biimtrid 152 |
. . 3
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23 | 3, 22 | relssdv 4736 |
. 2
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24 | ss0 3478 |
. 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 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-14 2163 ax-ext 2171 ax-sep 4136 ax-pow 4192 ax-pr 4227 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ral 2473 df-rex 2474 df-v 2754 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-nul 3438 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-br 4019 df-opab 4080 df-id 4311 df-po 4314 df-xp 4650 df-rel 4651 df-res 4656 |
This theorem is referenced by: (None) |
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