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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 6648 |
. 2
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2 | f1ofn 5376 |
. . . . 5
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3 | fndm 5230 |
. . . . . 6
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4 | vex 2692 |
. . . . . . 7
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5 | 4 | dmex 4813 |
. . . . . 6
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6 | 3, 5 | eqeltrrdi 2232 |
. . . . 5
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7 | 2, 6 | syl 14 |
. . . 4
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8 | f1ofo 5382 |
. . . . . 6
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9 | forn 5356 |
. . . . . 6
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10 | 8, 9 | syl 14 |
. . . . 5
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11 | 4 | rnex 4814 |
. . . . 5
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12 | 10, 11 | eqeltrrdi 2232 |
. . . 4
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13 | 7, 12 | jca 304 |
. . 3
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14 | 13 | exlimiv 1578 |
. 2
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15 | f1oeq2 5365 |
. . . 4
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16 | 15 | exbidv 1798 |
. . 3
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17 | f1oeq3 5366 |
. . . 4
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18 | 17 | exbidv 1798 |
. . 3
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19 | df-en 6643 |
. . 3
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20 | 16, 18, 19 | brabg 4199 |
. 2
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21 | 1, 14, 20 | pm5.21nii 694 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-13 1492 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-sep 4054 ax-pow 4106 ax-pr 4139 ax-un 4363 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ral 2422 df-rex 2423 df-v 2691 df-un 3080 df-in 3082 df-ss 3089 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-br 3938 df-opab 3998 df-xp 4553 df-rel 4554 df-cnv 4555 df-dm 4557 df-rn 4558 df-fn 5134 df-f 5135 df-f1 5136 df-fo 5137 df-f1o 5138 df-en 6643 |
This theorem is referenced by: domen 6653 f1oen3g 6656 ener 6681 en0 6697 ensn1 6698 en1 6701 unen 6718 enm 6722 xpen 6747 mapen 6748 ssenen 6753 phplem4 6757 phplem4on 6769 fidceq 6771 dif1en 6781 fin0 6787 fin0or 6788 en2eqpr 6809 fiintim 6825 fidcenumlemim 6848 enomnilem 7018 enmkvlem 7043 enwomnilem 7050 cc3 7100 hasheqf1o 10563 hashfacen 10611 fz1f1o 11176 ennnfonelemim 11973 exmidunben 11975 ctinfom 11977 qnnen 11980 enctlem 11981 ctiunct 11989 exmidsbthrlem 13392 sbthom 13396 |
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