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Mirrors > Home > MPE Home > Th. List > poirr2 | Structured version Visualization version 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 5138 |
. . . 4
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2 | relin2 4957 |
. . . 4
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3 | 1, 2 | mp1i 13 |
. . 3
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4 | df-br 4396 |
. . . . 5
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5 | brin 4445 |
. . . . 5
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6 | 4, 5 | bitr3i 259 |
. . . 4
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7 | vex 3034 |
. . . . . . . . 9
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8 | 7 | brres 5117 |
. . . . . . . 8
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9 | poirr 4771 |
. . . . . . . . . . 11
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10 | 7 | ideq 4992 |
. . . . . . . . . . . . 13
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11 | breq2 4399 |
. . . . . . . . . . . . 13
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12 | 10, 11 | sylbi 200 |
. . . . . . . . . . . 12
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13 | 12 | notbid 301 |
. . . . . . . . . . 11
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14 | 9, 13 | syl5ibcom 228 |
. . . . . . . . . 10
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15 | 14 | expimpd 614 |
. . . . . . . . 9
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16 | 15 | ancomsd 461 |
. . . . . . . 8
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17 | 8, 16 | syl5bi 225 |
. . . . . . 7
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18 | 17 | con2d 119 |
. . . . . 6
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19 | imnan 429 |
. . . . . 6
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20 | 18, 19 | sylib 201 |
. . . . 5
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21 | 20 | pm2.21d 109 |
. . . 4
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22 | 6, 21 | syl5bi 225 |
. . 3
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23 | 3, 22 | relssdv 4932 |
. 2
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24 | ss0 3768 |
. 2
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
25 | 23, 24 | syl 17 |
1
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Colors of variables: wff setvar class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1677 ax-4 1690 ax-5 1766 ax-6 1813 ax-7 1859 ax-9 1913 ax-10 1932 ax-11 1937 ax-12 1950 ax-13 2104 ax-ext 2451 ax-sep 4518 ax-nul 4527 ax-pr 4639 |
This theorem depends on definitions: df-bi 190 df-or 377 df-an 378 df-3an 1009 df-tru 1455 df-ex 1672 df-nf 1676 df-sb 1806 df-eu 2323 df-mo 2324 df-clab 2458 df-cleq 2464 df-clel 2467 df-nfc 2601 df-ne 2643 df-ral 2761 df-rex 2762 df-rab 2765 df-v 3033 df-dif 3393 df-un 3395 df-in 3397 df-ss 3404 df-nul 3723 df-if 3873 df-sn 3960 df-pr 3962 df-op 3966 df-br 4396 df-opab 4455 df-id 4754 df-po 4760 df-xp 4845 df-rel 4846 df-res 4851 |
This theorem is referenced by: (None) |
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