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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 6691 | . . . 4 | |
2 | 1 | a1i 9 | . . 3 |
3 | bren 6694 | . . . . 5 | |
4 | f1ocnv 5429 | . . . . . . 7 | |
5 | vex 2715 | . . . . . . . 8 | |
6 | vex 2715 | . . . . . . . 8 | |
7 | f1oen2g 6702 | . . . . . . . 8 | |
8 | 5, 6, 7 | mp3an12 1309 | . . . . . . 7 |
9 | 4, 8 | syl 14 | . . . . . 6 |
10 | 9 | exlimiv 1578 | . . . . 5 |
11 | 3, 10 | sylbi 120 | . . . 4 |
12 | 11 | adantl 275 | . . 3 |
13 | bren 6694 | . . . . 5 | |
14 | bren 6694 | . . . . 5 | |
15 | eeanv 1912 | . . . . . 6 | |
16 | f1oco 5439 | . . . . . . . . 9 | |
17 | 16 | ancoms 266 | . . . . . . . 8 |
18 | vex 2715 | . . . . . . . . 9 | |
19 | f1oen2g 6702 | . . . . . . . . 9 | |
20 | 6, 18, 19 | mp3an12 1309 | . . . . . . . 8 |
21 | 17, 20 | syl 14 | . . . . . . 7 |
22 | 21 | exlimivv 1876 | . . . . . 6 |
23 | 15, 22 | sylbir 134 | . . . . 5 |
24 | 13, 14, 23 | syl2anb 289 | . . . 4 |
25 | 24 | adantl 275 | . . 3 |
26 | 6 | enref 6712 | . . . . 5 |
27 | 6, 26 | 2th 173 | . . . 4 |
28 | 27 | a1i 9 | . . 3 |
29 | 2, 12, 25, 28 | iserd 6508 | . 2 |
30 | 29 | mptru 1344 | 1 |
Colors of variables: wff set class |
Syntax hints: wa 103 wb 104 wtru 1336 wex 1472 wcel 2128 cvv 2712 class class class wbr 3967 ccnv 4587 ccom 4592 wrel 4593 wf1o 5171 wer 6479 cen 6685 |
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 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-13 2130 ax-14 2131 ax-ext 2139 ax-sep 4084 ax-pow 4137 ax-pr 4171 ax-un 4395 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1338 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ral 2440 df-rex 2441 df-v 2714 df-un 3106 df-in 3108 df-ss 3115 df-pw 3546 df-sn 3567 df-pr 3568 df-op 3570 df-uni 3775 df-br 3968 df-opab 4028 df-id 4255 df-xp 4594 df-rel 4595 df-cnv 4596 df-co 4597 df-dm 4598 df-rn 4599 df-res 4600 df-ima 4601 df-fun 5174 df-fn 5175 df-f 5176 df-f1 5177 df-fo 5178 df-f1o 5179 df-er 6482 df-en 6688 |
This theorem is referenced by: ensymb 6727 entr 6731 |
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