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Mirrors > Home > ILE Home > Th. List > en1 | Unicode version |
Description: A set is equinumerous to ordinal one iff it is a singleton. (Contributed by NM, 25-Jul-2004.) |
Ref | Expression |
---|---|
en1 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df1o2 6430 |
. . . . 5
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2 | 1 | breq2i 4012 |
. . . 4
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3 | bren 6747 |
. . . 4
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4 | 2, 3 | bitri 184 |
. . 3
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5 | f1ocnv 5475 |
. . . . 5
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6 | f1ofo 5469 |
. . . . . . . 8
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7 | forn 5442 |
. . . . . . . 8
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8 | 6, 7 | syl 14 |
. . . . . . 7
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9 | f1of 5462 |
. . . . . . . . . 10
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10 | 0ex 4131 |
. . . . . . . . . . . 12
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11 | 10 | fsn2 5691 |
. . . . . . . . . . 11
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12 | 11 | simprbi 275 |
. . . . . . . . . 10
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13 | 9, 12 | syl 14 |
. . . . . . . . 9
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14 | 13 | rneqd 4857 |
. . . . . . . 8
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15 | 10 | rnsnop 5110 |
. . . . . . . 8
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16 | 14, 15 | eqtrdi 2226 |
. . . . . . 7
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17 | 8, 16 | eqtr3d 2212 |
. . . . . 6
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18 | 5, 17 | syl 14 |
. . . . 5
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19 | f1ofn 5463 |
. . . . . . 7
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20 | 10 | snid 3624 |
. . . . . . 7
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21 | funfvex 5533 |
. . . . . . . 8
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22 | 21 | funfni 5317 |
. . . . . . 7
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23 | 19, 20, 22 | sylancl 413 |
. . . . . 6
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24 | sneq 3604 |
. . . . . . . 8
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25 | 24 | eqeq2d 2189 |
. . . . . . 7
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26 | 25 | spcegv 2826 |
. . . . . 6
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27 | 23, 26 | syl 14 |
. . . . 5
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28 | 5, 18, 27 | sylc 62 |
. . . 4
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29 | 28 | exlimiv 1598 |
. . 3
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30 | 4, 29 | sylbi 121 |
. 2
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31 | vex 2741 |
. . . . 5
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32 | 31 | ensn1 6796 |
. . . 4
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33 | breq1 4007 |
. . . 4
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34 | 32, 33 | mpbiri 168 |
. . 3
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35 | 34 | exlimiv 1598 |
. 2
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36 | 30, 35 | impbii 126 |
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-in1 614 ax-in2 615 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-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4122 ax-nul 4130 ax-pow 4175 ax-pr 4210 ax-un 4434 |
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-reu 2462 df-v 2740 df-sbc 2964 df-dif 3132 df-un 3134 df-in 3136 df-ss 3143 df-nul 3424 df-pw 3578 df-sn 3599 df-pr 3600 df-op 3602 df-uni 3811 df-br 4005 df-opab 4066 df-id 4294 df-suc 4372 df-xp 4633 df-rel 4634 df-cnv 4635 df-co 4636 df-dm 4637 df-rn 4638 df-res 4639 df-ima 4640 df-iota 5179 df-fun 5219 df-fn 5220 df-f 5221 df-f1 5222 df-fo 5223 df-f1o 5224 df-fv 5225 df-1o 6417 df-en 6741 |
This theorem is referenced by: en1bg 6800 reuen1 6801 pm54.43 7189 |
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