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Mirrors > Home > ILE Home > Th. List > cnvsom | Unicode version |
Description: The converse of a strict order relation is a strict order relation. (Contributed by Jim Kingdon, 19-Dec-2018.) |
Ref | Expression |
---|---|
cnvsom |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cnvpom 4803 |
. . 3
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2 | vex 2554 |
. . . . . . . . 9
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3 | vex 2554 |
. . . . . . . . 9
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4 | 2, 3 | brcnv 4461 |
. . . . . . . 8
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5 | vex 2554 |
. . . . . . . . . . 11
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6 | 2, 5 | brcnv 4461 |
. . . . . . . . . 10
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7 | 5, 3 | brcnv 4461 |
. . . . . . . . . 10
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8 | 6, 7 | orbi12i 680 |
. . . . . . . . 9
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9 | orcom 646 |
. . . . . . . . 9
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10 | 8, 9 | bitri 173 |
. . . . . . . 8
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11 | 4, 10 | imbi12i 228 |
. . . . . . 7
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12 | 11 | ralbii 2324 |
. . . . . 6
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13 | 12 | 2ralbii 2326 |
. . . . 5
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14 | ralcom 2467 |
. . . . 5
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15 | 13, 14 | bitr3i 175 |
. . . 4
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16 | 15 | a1i 9 |
. . 3
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17 | 1, 16 | anbi12d 442 |
. 2
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18 | df-iso 4025 |
. 2
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19 | df-iso 4025 |
. 2
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20 | 17, 18, 19 | 3bitr4g 212 |
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-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-po 4024 df-iso 4025 df-cnv 4296 |
This theorem is referenced by: gtso 6894 |
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