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Mirrors > Home > ILE Home > Th. List > brcogw | Unicode version |
Description: Ordered pair membership in a composition. (Contributed by Thierry Arnoux, 14-Jan-2018.) |
Ref | Expression |
---|---|
brcogw |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl1 907 | . 2 | |
2 | simpl2 908 | . 2 | |
3 | breq2 3768 | . . . . . 6 | |
4 | breq1 3767 | . . . . . 6 | |
5 | 3, 4 | anbi12d 442 | . . . . 5 |
6 | 5 | spcegv 2641 | . . . 4 |
7 | 6 | imp 115 | . . 3 |
8 | 7 | 3ad2antl3 1068 | . 2 |
9 | brcog 4502 | . . 3 | |
10 | 9 | biimpar 281 | . 2 |
11 | 1, 2, 8, 10 | syl21anc 1134 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 97 w3a 885 wceq 1243 wex 1381 wcel 1393 class class class wbr 3764 ccom 4349 |
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 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-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-co 4354 |
This theorem is referenced by: (None) |
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