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Mirrors > Home > ILE Home > Th. List > relcnvtr | Unicode version |
Description: A relation is transitive iff its converse is transitive. (Contributed by FL, 19-Sep-2011.) |
Ref | Expression |
---|---|
relcnvtr |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cnvco 4463 |
. . 3
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2 | cnvss 4451 |
. . 3
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3 | 1, 2 | syl5eqssr 2984 |
. 2
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4 | cnvco 4463 |
. . . 4
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5 | cnvss 4451 |
. . . 4
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6 | sseq1 2960 |
. . . . 5
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7 | dfrel2 4714 |
. . . . . . 7
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8 | coeq1 4436 |
. . . . . . . . . 10
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9 | coeq2 4437 |
. . . . . . . . . 10
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10 | 8, 9 | eqtrd 2069 |
. . . . . . . . 9
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11 | id 19 |
. . . . . . . . 9
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12 | 10, 11 | sseq12d 2968 |
. . . . . . . 8
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13 | 12 | biimpd 132 |
. . . . . . 7
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14 | 7, 13 | sylbi 114 |
. . . . . 6
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15 | 14 | com12 27 |
. . . . 5
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16 | 6, 15 | syl6bi 152 |
. . . 4
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17 | 4, 5, 16 | mpsyl 59 |
. . 3
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18 | 17 | com12 27 |
. 2
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19 | 3, 18 | impbid2 131 |
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-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-un 2916 df-in 2918 df-ss 2925 df-pw 3353 df-sn 3373 df-pr 3374 df-op 3376 df-br 3756 df-opab 3810 df-xp 4294 df-rel 4295 df-cnv 4296 df-co 4297 |
This theorem is referenced by: (None) |
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