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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 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | encv 6463 |
. 2
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2 | f1ofn 5254 |
. . . . 5
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3 | fndm 5113 |
. . . . . 6
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4 | vex 2622 |
. . . . . . 7
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5 | 4 | dmex 4699 |
. . . . . 6
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6 | 3, 5 | syl6eqelr 2179 |
. . . . 5
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7 | 2, 6 | syl 14 |
. . . 4
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8 | f1ofo 5260 |
. . . . . 6
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9 | forn 5236 |
. . . . . 6
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10 | 8, 9 | syl 14 |
. . . . 5
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11 | 4 | rnex 4700 |
. . . . 5
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12 | 10, 11 | syl6eqelr 2179 |
. . . 4
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13 | 7, 12 | jca 300 |
. . 3
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14 | 13 | exlimiv 1534 |
. 2
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15 | f1oeq2 5245 |
. . . 4
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16 | 15 | exbidv 1753 |
. . 3
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17 | f1oeq3 5246 |
. . . 4
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18 | 17 | exbidv 1753 |
. . 3
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19 | df-en 6458 |
. . 3
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20 | 16, 18, 19 | brabg 4096 |
. 2
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21 | 1, 14, 20 | pm5.21nii 655 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 665 ax-5 1381 ax-7 1382 ax-gen 1383 ax-ie1 1427 ax-ie2 1428 ax-8 1440 ax-10 1441 ax-11 1442 ax-i12 1443 ax-bndl 1444 ax-4 1445 ax-13 1449 ax-14 1450 ax-17 1464 ax-i9 1468 ax-ial 1472 ax-i5r 1473 ax-ext 2070 ax-sep 3957 ax-pow 4009 ax-pr 4036 ax-un 4260 |
This theorem depends on definitions: df-bi 115 df-3an 926 df-tru 1292 df-nf 1395 df-sb 1693 df-eu 1951 df-mo 1952 df-clab 2075 df-cleq 2081 df-clel 2084 df-nfc 2217 df-ral 2364 df-rex 2365 df-v 2621 df-un 3003 df-in 3005 df-ss 3012 df-pw 3431 df-sn 3452 df-pr 3453 df-op 3455 df-uni 3654 df-br 3846 df-opab 3900 df-xp 4444 df-rel 4445 df-cnv 4446 df-dm 4448 df-rn 4449 df-fn 5018 df-f 5019 df-f1 5020 df-fo 5021 df-f1o 5022 df-en 6458 |
This theorem is referenced by: domen 6468 f1oen3g 6471 ener 6496 en0 6512 ensn1 6513 en1 6516 unen 6533 enm 6536 xpen 6561 mapen 6562 ssenen 6567 phplem4 6571 phplem4on 6583 fidceq 6585 dif1en 6595 fin0 6601 fin0or 6602 en2eqpr 6623 fiintim 6639 fidcenumlemim 6661 enomnilem 6794 hasheqf1o 10193 hashfacen 10241 fz1f1o 10764 exmidsbthrlem 11912 |
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