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Mirrors > Home > ILE Home > Th. List > f1osng | Unicode version |
Description: A singleton of an ordered pair is one-to-one onto function. (Contributed by Mario Carneiro, 12-Jan-2013.) |
Ref | Expression |
---|---|
f1osng |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sneq 3603 |
. . . 4
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2 | f1oeq2 5450 |
. . . 4
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3 | 1, 2 | syl 14 |
. . 3
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4 | opeq1 3778 |
. . . . 5
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5 | 4 | sneqd 3605 |
. . . 4
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6 | f1oeq1 5449 |
. . . 4
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7 | 5, 6 | syl 14 |
. . 3
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8 | 3, 7 | bitrd 188 |
. 2
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9 | sneq 3603 |
. . . 4
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10 | f1oeq3 5451 |
. . . 4
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11 | 9, 10 | syl 14 |
. . 3
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12 | opeq2 3779 |
. . . . 5
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13 | 12 | sneqd 3605 |
. . . 4
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14 | f1oeq1 5449 |
. . . 4
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15 | 13, 14 | syl 14 |
. . 3
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16 | 11, 15 | bitrd 188 |
. 2
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17 | vex 2740 |
. . 3
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18 | vex 2740 |
. . 3
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19 | 17, 18 | f1osn 5501 |
. 2
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20 | 8, 16, 19 | vtocl2g 2801 |
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 106 ax-ia2 107 ax-ia3 108 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-14 2151 ax-ext 2159 ax-sep 4121 ax-pow 4174 ax-pr 4209 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ral 2460 df-rex 2461 df-v 2739 df-un 3133 df-in 3135 df-ss 3142 df-pw 3577 df-sn 3598 df-pr 3599 df-op 3601 df-br 4004 df-opab 4065 df-id 4293 df-xp 4632 df-rel 4633 df-cnv 4634 df-co 4635 df-dm 4636 df-rn 4637 df-fun 5218 df-fn 5219 df-f 5220 df-f1 5221 df-fo 5222 df-f1o 5223 |
This theorem is referenced by: f1sng 5503 f1oprg 5505 fsnunf 5716 dif1en 6878 1fv 10138 zfz1isolem1 10819 sumsnf 11416 prodsnf 11599 ennnfonelemhf1o 12413 |
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