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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 4582 |
. . . 4
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2 | relin2 4399 |
. . . 4
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3 | 1, 2 | mp1i 10 |
. . 3
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4 | df-br 3756 |
. . . . 5
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5 | brin 3802 |
. . . . 5
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6 | 4, 5 | bitr3i 175 |
. . . 4
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7 | vex 2554 |
. . . . . . . . 9
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8 | 7 | brres 4561 |
. . . . . . . 8
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9 | poirr 4035 |
. . . . . . . . . . 11
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10 | 7 | ideq 4431 |
. . . . . . . . . . . . 13
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11 | breq2 3759 |
. . . . . . . . . . . . 13
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12 | 10, 11 | sylbi 114 |
. . . . . . . . . . . 12
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13 | 12 | notbid 591 |
. . . . . . . . . . 11
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14 | 9, 13 | syl5ibcom 144 |
. . . . . . . . . 10
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15 | 14 | expimpd 345 |
. . . . . . . . 9
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16 | 15 | ancomsd 256 |
. . . . . . . 8
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17 | 8, 16 | syl5bi 141 |
. . . . . . 7
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18 | 17 | con2d 554 |
. . . . . 6
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19 | imnan 623 |
. . . . . 6
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20 | 18, 19 | sylib 127 |
. . . . 5
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21 | 20 | pm2.21d 549 |
. . . 4
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22 | 6, 21 | syl5bi 141 |
. . 3
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23 | 3, 22 | relssdv 4375 |
. 2
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24 | ss0 3251 |
. 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-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-14 1402 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 ax-sep 3866 ax-pow 3918 ax-pr 3935 |
This theorem depends on definitions: df-bi 110 df-3an 886 df-tru 1245 df-nf 1347 df-sb 1643 df-eu 1900 df-mo 1901 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-ral 2305 df-rex 2306 df-v 2553 df-dif 2914 df-un 2916 df-in 2918 df-ss 2925 df-nul 3219 df-pw 3353 df-sn 3373 df-pr 3374 df-op 3376 df-br 3756 df-opab 3810 df-id 4021 df-po 4024 df-xp 4294 df-rel 4295 df-res 4300 |
This theorem is referenced by: (None) |
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