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Mirrors > Home > ILE Home > Th. List > 0er | Unicode version |
Description: The empty set is an equivalence relation on the empty set. (Contributed by Mario Carneiro, 5-Sep-2015.) |
Ref | Expression |
---|---|
0er |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rel0 4480 | . . . 4 | |
2 | 1 | a1i 9 | . . 3 |
3 | df-br 3786 | . . . . 5 | |
4 | noel 3255 | . . . . . 6 | |
5 | 4 | pm2.21i 607 | . . . . 5 |
6 | 3, 5 | sylbi 119 | . . . 4 |
7 | 6 | adantl 271 | . . 3 |
8 | 4 | pm2.21i 607 | . . . . 5 |
9 | 3, 8 | sylbi 119 | . . . 4 |
10 | 9 | ad2antrl 473 | . . 3 |
11 | noel 3255 | . . . . . 6 | |
12 | noel 3255 | . . . . . 6 | |
13 | 11, 12 | 2false 649 | . . . . 5 |
14 | df-br 3786 | . . . . 5 | |
15 | 13, 14 | bitr4i 185 | . . . 4 |
16 | 15 | a1i 9 | . . 3 |
17 | 2, 7, 10, 16 | iserd 6155 | . 2 |
18 | 17 | trud 1293 | 1 |
Colors of variables: wff set class |
Syntax hints: wb 103 wtru 1285 wcel 1433 c0 3251 cop 3401 class class class wbr 3785 wrel 4368 wer 6126 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-pow 3948 ax-pr 3964 |
This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ral 2353 df-rex 2354 df-v 2603 df-dif 2975 df-un 2977 df-in 2979 df-ss 2986 df-nul 3252 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-br 3786 df-opab 3840 df-xp 4369 df-rel 4370 df-cnv 4371 df-co 4372 df-dm 4373 df-er 6129 |
This theorem is referenced by: (None) |
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