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Mirrors > Home > ILE Home > Th. List > f1o00 | Unicode version |
Description: One-to-one onto mapping of the empty set. (Contributed by NM, 15-Apr-1998.) |
Ref | Expression |
---|---|
f1o00 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dff1o4 5261 |
. 2
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2 | fn0 5133 |
. . . . . 6
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3 | 2 | biimpi 118 |
. . . . 5
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4 | 3 | adantr 270 |
. . . 4
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5 | dm0 4650 |
. . . . 5
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6 | cnveq 4610 |
. . . . . . . . . 10
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7 | cnv0 4835 |
. . . . . . . . . 10
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8 | 6, 7 | syl6eq 2136 |
. . . . . . . . 9
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9 | 2, 8 | sylbi 119 |
. . . . . . . 8
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10 | 9 | fneq1d 5104 |
. . . . . . 7
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11 | 10 | biimpa 290 |
. . . . . 6
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12 | fndm 5113 |
. . . . . 6
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13 | 11, 12 | syl 14 |
. . . . 5
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14 | 5, 13 | syl5reqr 2135 |
. . . 4
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15 | 4, 14 | jca 300 |
. . 3
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16 | 2 | biimpri 131 |
. . . . 5
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17 | 16 | adantr 270 |
. . . 4
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18 | eqid 2088 |
. . . . . 6
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19 | fn0 5133 |
. . . . . 6
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20 | 18, 19 | mpbir 144 |
. . . . 5
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21 | 8 | fneq1d 5104 |
. . . . . 6
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22 | fneq2 5103 |
. . . . . 6
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23 | 21, 22 | sylan9bb 450 |
. . . . 5
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24 | 20, 23 | mpbiri 166 |
. . . 4
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25 | 17, 24 | jca 300 |
. . 3
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26 | 15, 25 | impbii 124 |
. 2
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27 | 1, 26 | bitri 182 |
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-in1 579 ax-in2 580 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-14 1450 ax-17 1464 ax-i9 1468 ax-ial 1472 ax-i5r 1473 ax-ext 2070 ax-sep 3957 ax-nul 3965 ax-pow 4009 ax-pr 4036 |
This theorem depends on definitions: df-bi 115 df-3an 926 df-tru 1292 df-fal 1295 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-dif 3001 df-un 3003 df-in 3005 df-ss 3012 df-nul 3287 df-pw 3431 df-sn 3452 df-pr 3453 df-op 3455 df-br 3846 df-opab 3900 df-id 4120 df-xp 4444 df-rel 4445 df-cnv 4446 df-co 4447 df-dm 4448 df-rn 4449 df-fun 5017 df-fn 5018 df-f 5019 df-f1 5020 df-fo 5021 df-f1o 5022 |
This theorem is referenced by: fo00 5289 f1o0 5290 en0 6512 |
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