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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 6638 | . . . 4 | |
2 | 1 | a1i 9 | . . 3 |
3 | bren 6641 | . . . . 5 | |
4 | f1ocnv 5380 | . . . . . . 7 | |
5 | vex 2689 | . . . . . . . 8 | |
6 | vex 2689 | . . . . . . . 8 | |
7 | f1oen2g 6649 | . . . . . . . 8 | |
8 | 5, 6, 7 | mp3an12 1305 | . . . . . . 7 |
9 | 4, 8 | syl 14 | . . . . . 6 |
10 | 9 | exlimiv 1577 | . . . . 5 |
11 | 3, 10 | sylbi 120 | . . . 4 |
12 | 11 | adantl 275 | . . 3 |
13 | bren 6641 | . . . . 5 | |
14 | bren 6641 | . . . . 5 | |
15 | eeanv 1904 | . . . . . 6 | |
16 | f1oco 5390 | . . . . . . . . 9 | |
17 | 16 | ancoms 266 | . . . . . . . 8 |
18 | vex 2689 | . . . . . . . . 9 | |
19 | f1oen2g 6649 | . . . . . . . . 9 | |
20 | 6, 18, 19 | mp3an12 1305 | . . . . . . . 8 |
21 | 17, 20 | syl 14 | . . . . . . 7 |
22 | 21 | exlimivv 1868 | . . . . . 6 |
23 | 15, 22 | sylbir 134 | . . . . 5 |
24 | 13, 14, 23 | syl2anb 289 | . . . 4 |
25 | 24 | adantl 275 | . . 3 |
26 | 6 | enref 6659 | . . . . 5 |
27 | 6, 26 | 2th 173 | . . . 4 |
28 | 27 | a1i 9 | . . 3 |
29 | 2, 12, 25, 28 | iserd 6455 | . 2 |
30 | 29 | mptru 1340 | 1 |
Colors of variables: wff set class |
Syntax hints: wa 103 wb 104 wtru 1332 wex 1468 wcel 1480 cvv 2686 class class class wbr 3929 ccnv 4538 ccom 4543 wrel 4544 wf1o 5122 wer 6426 cen 6632 |
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-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-13 1491 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 ax-un 4355 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 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-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-br 3930 df-opab 3990 df-id 4215 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-fun 5125 df-fn 5126 df-f 5127 df-f1 5128 df-fo 5129 df-f1o 5130 df-er 6429 df-en 6635 |
This theorem is referenced by: ensymb 6674 entr 6678 |
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