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Mirrors > Home > ILE Home > Th. List > bren | Unicode version |
Description: Equinumerosity relation. (Contributed by NM, 15-Jun-1998.) |
Ref | Expression |
---|---|
bren |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | encv 6640 | . 2 | |
2 | f1ofn 5368 | . . . . 5 | |
3 | fndm 5222 | . . . . . 6 | |
4 | vex 2689 | . . . . . . 7 | |
5 | 4 | dmex 4805 | . . . . . 6 |
6 | 3, 5 | eqeltrrdi 2231 | . . . . 5 |
7 | 2, 6 | syl 14 | . . . 4 |
8 | f1ofo 5374 | . . . . . 6 | |
9 | forn 5348 | . . . . . 6 | |
10 | 8, 9 | syl 14 | . . . . 5 |
11 | 4 | rnex 4806 | . . . . 5 |
12 | 10, 11 | eqeltrrdi 2231 | . . . 4 |
13 | 7, 12 | jca 304 | . . 3 |
14 | 13 | exlimiv 1577 | . 2 |
15 | f1oeq2 5357 | . . . 4 | |
16 | 15 | exbidv 1797 | . . 3 |
17 | f1oeq3 5358 | . . . 4 | |
18 | 17 | exbidv 1797 | . . 3 |
19 | df-en 6635 | . . 3 | |
20 | 16, 18, 19 | brabg 4191 | . 2 |
21 | 1, 14, 20 | pm5.21nii 693 | 1 |
Colors of variables: wff set class |
Syntax hints: wa 103 wb 104 wceq 1331 wex 1468 wcel 1480 cvv 2686 class class class wbr 3929 cdm 4539 crn 4540 wfn 5118 wfo 5121 wf1o 5122 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-xp 4545 df-rel 4546 df-cnv 4547 df-dm 4549 df-rn 4550 df-fn 5126 df-f 5127 df-f1 5128 df-fo 5129 df-f1o 5130 df-en 6635 |
This theorem is referenced by: domen 6645 f1oen3g 6648 ener 6673 en0 6689 ensn1 6690 en1 6693 unen 6710 enm 6714 xpen 6739 mapen 6740 ssenen 6745 phplem4 6749 phplem4on 6761 fidceq 6763 dif1en 6773 fin0 6779 fin0or 6780 en2eqpr 6801 fiintim 6817 fidcenumlemim 6840 enomnilem 7010 hasheqf1o 10531 hashfacen 10579 fz1f1o 11144 ennnfonelemim 11937 exmidunben 11939 ctinfom 11941 qnnen 11944 enctlem 11945 ctiunct 11953 exmidsbthrlem 13217 sbthom 13221 |
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