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Mirrors > Home > MPE Home > Th. List > cotr | Unicode version |
Description: Two ways of saying a relation is transitive. Definition of transitivity in [Schechter] p. 51. (Contributed by NM, 27-Dec-1996.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) |
Ref | Expression |
---|---|
cotr |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-co 4846 |
. . . 4
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2 | 1 | relopabi 4959 |
. . 3
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3 | ssrel 4923 |
. . 3
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4 | 2, 3 | ax-mp 8 |
. 2
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5 | vex 2919 |
. . . . . . . 8
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6 | vex 2919 |
. . . . . . . 8
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7 | 5, 6 | opelco 5003 |
. . . . . . 7
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8 | df-br 4173 |
. . . . . . . 8
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9 | 8 | bicomi 194 |
. . . . . . 7
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10 | 7, 9 | imbi12i 317 |
. . . . . 6
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11 | 19.23v 1910 |
. . . . . 6
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12 | 10, 11 | bitr4i 244 |
. . . . 5
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13 | 12 | albii 1572 |
. . . 4
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14 | alcom 1748 |
. . . 4
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15 | 13, 14 | bitri 241 |
. . 3
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16 | 15 | albii 1572 |
. 2
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17 | 4, 16 | bitri 241 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem is referenced by: xpidtr 5215 trin2 5216 dfer2 6865 pslem 14593 letsr 14627 dirtr 14636 filnetlem3 26299 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-3 7 ax-mp 8 ax-gen 1552 ax-5 1563 ax-17 1623 ax-9 1662 ax-8 1683 ax-14 1725 ax-6 1740 ax-7 1745 ax-11 1757 ax-12 1946 ax-ext 2385 ax-sep 4290 ax-nul 4298 ax-pr 4363 |
This theorem depends on definitions: df-bi 178 df-or 360 df-an 361 df-3an 938 df-tru 1325 df-ex 1548 df-nf 1551 df-sb 1656 df-eu 2258 df-mo 2259 df-clab 2391 df-cleq 2397 df-clel 2400 df-nfc 2529 df-ne 2569 df-ral 2671 df-rex 2672 df-rab 2675 df-v 2918 df-dif 3283 df-un 3285 df-in 3287 df-ss 3294 df-nul 3589 df-if 3700 df-sn 3780 df-pr 3781 df-op 3783 df-br 4173 df-opab 4227 df-xp 4843 df-rel 4844 df-co 4846 |
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