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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 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cnvpom 4860 | . . 3 | |
2 | vex 2560 | . . . . . . . . 9 | |
3 | vex 2560 | . . . . . . . . 9 | |
4 | 2, 3 | brcnv 4518 | . . . . . . . 8 |
5 | vex 2560 | . . . . . . . . . . 11 | |
6 | 2, 5 | brcnv 4518 | . . . . . . . . . 10 |
7 | 5, 3 | brcnv 4518 | . . . . . . . . . 10 |
8 | 6, 7 | orbi12i 681 | . . . . . . . . 9 |
9 | orcom 647 | . . . . . . . . 9 | |
10 | 8, 9 | bitri 173 | . . . . . . . 8 |
11 | 4, 10 | imbi12i 228 | . . . . . . 7 |
12 | 11 | ralbii 2330 | . . . . . 6 |
13 | 12 | 2ralbii 2332 | . . . . 5 |
14 | ralcom 2473 | . . . . 5 | |
15 | 13, 14 | bitr3i 175 | . . . 4 |
16 | 15 | a1i 9 | . . 3 |
17 | 1, 16 | anbi12d 442 | . 2 |
18 | df-iso 4034 | . 2 | |
19 | df-iso 4034 | . 2 | |
20 | 17, 18, 19 | 3bitr4g 212 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 97 wb 98 wo 629 wex 1381 wcel 1393 wral 2306 class class class wbr 3764 wpo 4031 wor 4032 ccnv 4344 |
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 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 ax-pow 3927 ax-pr 3944 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ral 2311 df-v 2559 df-un 2922 df-in 2924 df-ss 2931 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-br 3765 df-opab 3819 df-po 4033 df-iso 4034 df-cnv 4353 |
This theorem is referenced by: gtso 7097 |
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