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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 6608 | . 2 | |
2 | f1ofn 5336 | . . . . 5 | |
3 | fndm 5192 | . . . . . 6 | |
4 | vex 2663 | . . . . . . 7 | |
5 | 4 | dmex 4775 | . . . . . 6 |
6 | 3, 5 | syl6eqelr 2209 | . . . . 5 |
7 | 2, 6 | syl 14 | . . . 4 |
8 | f1ofo 5342 | . . . . . 6 | |
9 | forn 5318 | . . . . . 6 | |
10 | 8, 9 | syl 14 | . . . . 5 |
11 | 4 | rnex 4776 | . . . . 5 |
12 | 10, 11 | syl6eqelr 2209 | . . . 4 |
13 | 7, 12 | jca 304 | . . 3 |
14 | 13 | exlimiv 1562 | . 2 |
15 | f1oeq2 5327 | . . . 4 | |
16 | 15 | exbidv 1781 | . . 3 |
17 | f1oeq3 5328 | . . . 4 | |
18 | 17 | exbidv 1781 | . . 3 |
19 | df-en 6603 | . . 3 | |
20 | 16, 18, 19 | brabg 4161 | . 2 |
21 | 1, 14, 20 | pm5.21nii 678 | 1 |
Colors of variables: wff set class |
Syntax hints: wa 103 wb 104 wceq 1316 wex 1453 wcel 1465 cvv 2660 class class class wbr 3899 cdm 4509 crn 4510 wfn 5088 wfo 5091 wf1o 5092 cen 6600 |
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 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-13 1476 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-pow 4068 ax-pr 4101 ax-un 4325 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ral 2398 df-rex 2399 df-v 2662 df-un 3045 df-in 3047 df-ss 3054 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-br 3900 df-opab 3960 df-xp 4515 df-rel 4516 df-cnv 4517 df-dm 4519 df-rn 4520 df-fn 5096 df-f 5097 df-f1 5098 df-fo 5099 df-f1o 5100 df-en 6603 |
This theorem is referenced by: domen 6613 f1oen3g 6616 ener 6641 en0 6657 ensn1 6658 en1 6661 unen 6678 enm 6682 xpen 6707 mapen 6708 ssenen 6713 phplem4 6717 phplem4on 6729 fidceq 6731 dif1en 6741 fin0 6747 fin0or 6748 en2eqpr 6769 fiintim 6785 fidcenumlemim 6808 enomnilem 6978 hasheqf1o 10499 hashfacen 10547 fz1f1o 11112 ennnfonelemim 11864 exmidunben 11866 ctinfom 11868 qnnen 11871 enctlem 11872 ctiunct 11880 exmidsbthrlem 13144 sbthom 13148 |
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