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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 6194 |
. . . . 5
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2 | 1 | breq2i 3853 |
. . . 4
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3 | bren 6464 |
. . . 4
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4 | 2, 3 | bitri 182 |
. . 3
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5 | f1ocnv 5266 |
. . . . 5
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6 | f1ofo 5260 |
. . . . . . . 8
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7 | forn 5236 |
. . . . . . . 8
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8 | 6, 7 | syl 14 |
. . . . . . 7
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9 | f1of 5253 |
. . . . . . . . . 10
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10 | 0ex 3966 |
. . . . . . . . . . . 12
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11 | 10 | fsn2 5471 |
. . . . . . . . . . 11
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12 | 11 | simprbi 269 |
. . . . . . . . . 10
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13 | 9, 12 | syl 14 |
. . . . . . . . 9
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14 | 13 | rneqd 4664 |
. . . . . . . 8
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15 | 10 | rnsnop 4911 |
. . . . . . . 8
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16 | 14, 15 | syl6eq 2136 |
. . . . . . 7
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17 | 8, 16 | eqtr3d 2122 |
. . . . . 6
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18 | 5, 17 | syl 14 |
. . . . 5
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19 | f1ofn 5254 |
. . . . . . 7
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20 | 10 | snid 3475 |
. . . . . . 7
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21 | funfvex 5322 |
. . . . . . . 8
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22 | 21 | funfni 5114 |
. . . . . . 7
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23 | 19, 20, 22 | sylancl 404 |
. . . . . 6
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24 | sneq 3457 |
. . . . . . . 8
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25 | 24 | eqeq2d 2099 |
. . . . . . 7
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26 | 25 | spcegv 2707 |
. . . . . 6
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27 | 23, 26 | syl 14 |
. . . . 5
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28 | 5, 18, 27 | sylc 61 |
. . . 4
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29 | 28 | exlimiv 1534 |
. . 3
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30 | 4, 29 | sylbi 119 |
. 2
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31 | vex 2622 |
. . . . 5
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32 | 31 | ensn1 6513 |
. . . 4
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33 | breq1 3848 |
. . . 4
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34 | 32, 33 | mpbiri 166 |
. . 3
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35 | 34 | exlimiv 1534 |
. 2
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36 | 30, 35 | impbii 124 |
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-13 1449 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 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-reu 2366 df-v 2621 df-sbc 2841 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-uni 3654 df-br 3846 df-opab 3900 df-id 4120 df-suc 4198 df-xp 4444 df-rel 4445 df-cnv 4446 df-co 4447 df-dm 4448 df-rn 4449 df-res 4450 df-ima 4451 df-iota 4980 df-fun 5017 df-fn 5018 df-f 5019 df-f1 5020 df-fo 5021 df-f1o 5022 df-fv 5023 df-1o 6181 df-en 6458 |
This theorem is referenced by: en1bg 6517 reuen1 6518 pm54.43 6818 |
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