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Mirrors > Home > ILE Home > Th. List > co01 | Unicode version |
Description: Composition with the empty set. (Contributed by NM, 24-Apr-2004.) |
Ref | Expression |
---|---|
co01 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cnv0 4942 | . . . 4 | |
2 | cnvco 4724 | . . . . 5 | |
3 | 1 | coeq2i 4699 | . . . . 5 |
4 | co02 5052 | . . . . 5 | |
5 | 2, 3, 4 | 3eqtri 2164 | . . . 4 |
6 | 1, 5 | eqtr4i 2163 | . . 3 |
7 | 6 | cnveqi 4714 | . 2 |
8 | rel0 4664 | . . 3 | |
9 | dfrel2 4989 | . . 3 | |
10 | 8, 9 | mpbi 144 | . 2 |
11 | relco 5037 | . . 3 | |
12 | dfrel2 4989 | . . 3 | |
13 | 11, 12 | mpbi 144 | . 2 |
14 | 7, 10, 13 | 3eqtr3ri 2169 | 1 |
Colors of variables: wff set class |
Syntax hints: wceq 1331 c0 3363 ccnv 4538 ccom 4543 wrel 4544 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-rex 2422 df-v 2688 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-nul 3364 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-br 3930 df-opab 3990 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 |
This theorem is referenced by: (None) |
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