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Mirrors > Home > ILE Home > Th. List > ener | Unicode version |
Description: Equinumerosity is an equivalence relation. (Contributed by NM, 19-Mar-1998.) (Revised by Mario Carneiro, 15-Nov-2014.) |
Ref | Expression |
---|---|
ener |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | relen 6225 | . . . 4 | |
2 | 1 | a1i 9 | . . 3 |
3 | bren 6228 | . . . . 5 | |
4 | f1ocnv 5139 | . . . . . . 7 | |
5 | vex 2560 | . . . . . . . 8 | |
6 | vex 2560 | . . . . . . . 8 | |
7 | f1oen2g 6235 | . . . . . . . 8 | |
8 | 5, 6, 7 | mp3an12 1222 | . . . . . . 7 |
9 | 4, 8 | syl 14 | . . . . . 6 |
10 | 9 | exlimiv 1489 | . . . . 5 |
11 | 3, 10 | sylbi 114 | . . . 4 |
12 | 11 | adantl 262 | . . 3 |
13 | bren 6228 | . . . . 5 | |
14 | bren 6228 | . . . . 5 | |
15 | eeanv 1807 | . . . . . 6 | |
16 | f1oco 5149 | . . . . . . . . 9 | |
17 | 16 | ancoms 255 | . . . . . . . 8 |
18 | vex 2560 | . . . . . . . . 9 | |
19 | f1oen2g 6235 | . . . . . . . . 9 | |
20 | 6, 18, 19 | mp3an12 1222 | . . . . . . . 8 |
21 | 17, 20 | syl 14 | . . . . . . 7 |
22 | 21 | exlimivv 1776 | . . . . . 6 |
23 | 15, 22 | sylbir 125 | . . . . 5 |
24 | 13, 14, 23 | syl2anb 275 | . . . 4 |
25 | 24 | adantl 262 | . . 3 |
26 | 6 | enref 6245 | . . . . 5 |
27 | 6, 26 | 2th 163 | . . . 4 |
28 | 27 | a1i 9 | . . 3 |
29 | 2, 12, 25, 28 | iserd 6132 | . 2 |
30 | 29 | trud 1252 | 1 |
Colors of variables: wff set class |
Syntax hints: wa 97 wb 98 wtru 1244 wex 1381 wcel 1393 cvv 2557 class class class wbr 3764 ccnv 4344 ccom 4349 wrel 4350 wf1o 4901 wer 6103 cen 6219 |
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-13 1404 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 ax-un 4170 |
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-rex 2312 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-uni 3581 df-br 3765 df-opab 3819 df-id 4030 df-xp 4351 df-rel 4352 df-cnv 4353 df-co 4354 df-dm 4355 df-rn 4356 df-res 4357 df-ima 4358 df-fun 4904 df-fn 4905 df-f 4906 df-f1 4907 df-fo 4908 df-f1o 4909 df-er 6106 df-en 6222 |
This theorem is referenced by: ensymb 6260 entr 6264 |
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