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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 6208 |
. . . . 5
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2 | 1 | breq2i 3859 |
. . . 4
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3 | bren 6518 |
. . . 4
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4 | 2, 3 | bitri 183 |
. . 3
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5 | f1ocnv 5279 |
. . . . 5
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6 | f1ofo 5273 |
. . . . . . . 8
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7 | forn 5249 |
. . . . . . . 8
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8 | 6, 7 | syl 14 |
. . . . . . 7
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9 | f1of 5266 |
. . . . . . . . . 10
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10 | 0ex 3972 |
. . . . . . . . . . . 12
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11 | 10 | fsn2 5485 |
. . . . . . . . . . 11
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12 | 11 | simprbi 270 |
. . . . . . . . . 10
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13 | 9, 12 | syl 14 |
. . . . . . . . 9
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14 | 13 | rneqd 4677 |
. . . . . . . 8
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15 | 10 | rnsnop 4924 |
. . . . . . . 8
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16 | 14, 15 | syl6eq 2137 |
. . . . . . 7
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17 | 8, 16 | eqtr3d 2123 |
. . . . . 6
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18 | 5, 17 | syl 14 |
. . . . 5
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19 | f1ofn 5267 |
. . . . . . 7
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20 | 10 | snid 3479 |
. . . . . . 7
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21 | funfvex 5335 |
. . . . . . . 8
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22 | 21 | funfni 5127 |
. . . . . . 7
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23 | 19, 20, 22 | sylancl 405 |
. . . . . 6
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24 | sneq 3461 |
. . . . . . . 8
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25 | 24 | eqeq2d 2100 |
. . . . . . 7
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26 | 25 | spcegv 2708 |
. . . . . 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 1535 |
. . 3
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30 | 4, 29 | sylbi 120 |
. 2
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31 | vex 2623 |
. . . . 5
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32 | 31 | ensn1 6567 |
. . . 4
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33 | breq1 3854 |
. . . 4
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34 | 32, 33 | mpbiri 167 |
. . 3
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35 | 34 | exlimiv 1535 |
. 2
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36 | 30, 35 | impbii 125 |
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 105 ax-ia2 106 ax-ia3 107 ax-in1 580 ax-in2 581 ax-io 666 ax-5 1382 ax-7 1383 ax-gen 1384 ax-ie1 1428 ax-ie2 1429 ax-8 1441 ax-10 1442 ax-11 1443 ax-i12 1444 ax-bndl 1445 ax-4 1446 ax-13 1450 ax-14 1451 ax-17 1465 ax-i9 1469 ax-ial 1473 ax-i5r 1474 ax-ext 2071 ax-sep 3963 ax-nul 3971 ax-pow 4015 ax-pr 4045 ax-un 4269 |
This theorem depends on definitions: df-bi 116 df-3an 927 df-tru 1293 df-nf 1396 df-sb 1694 df-eu 1952 df-mo 1953 df-clab 2076 df-cleq 2082 df-clel 2085 df-nfc 2218 df-ral 2365 df-rex 2366 df-reu 2367 df-v 2622 df-sbc 2842 df-dif 3002 df-un 3004 df-in 3006 df-ss 3013 df-nul 3288 df-pw 3435 df-sn 3456 df-pr 3457 df-op 3459 df-uni 3660 df-br 3852 df-opab 3906 df-id 4129 df-suc 4207 df-xp 4458 df-rel 4459 df-cnv 4460 df-co 4461 df-dm 4462 df-rn 4463 df-res 4464 df-ima 4465 df-iota 4993 df-fun 5030 df-fn 5031 df-f 5032 df-f1 5033 df-fo 5034 df-f1o 5035 df-fv 5036 df-1o 6195 df-en 6512 |
This theorem is referenced by: en1bg 6571 reuen1 6572 pm54.43 6879 |
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