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Mirrors > Home > MPE Home > Th. List > opthwiener | Structured version Visualization version Unicode version |
Description: Justification theorem for the ordered pair definition in Norbert Wiener, "A simplification of the logic of relations," Proc. of the Cambridge Philos. Soc., 1914, vol. 17, pp.387-390. It is also shown as a definition in [Enderton] p. 36 and as Exercise 4.8(b) of [Mendelson] p. 230. It is meaningful only for classes that exist as sets (i.e. are not proper classes). See df-op 3974 for other ordered pair definitions. (Contributed by NM, 28-Sep-2003.) |
Ref | Expression |
---|---|
opthw.1 |
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opthw.2 |
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Ref | Expression |
---|---|
opthwiener |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | id 22 |
. . . . . . 7
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2 | snex 4640 |
. . . . . . . . . . . 12
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3 | 2 | prid2 4080 |
. . . . . . . . . . 11
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4 | eleq2 2517 |
. . . . . . . . . . 11
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5 | 3, 4 | mpbii 215 |
. . . . . . . . . 10
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6 | 2 | elpr 3985 |
. . . . . . . . . 10
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7 | 5, 6 | sylib 200 |
. . . . . . . . 9
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8 | 0ex 4534 |
. . . . . . . . . . . . 13
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9 | 8 | prid2 4080 |
. . . . . . . . . . . 12
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10 | opthw.2 |
. . . . . . . . . . . . . 14
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11 | 10 | snnz 4089 |
. . . . . . . . . . . . 13
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12 | 8 | elsnc 3991 |
. . . . . . . . . . . . . 14
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13 | eqcom 2457 |
. . . . . . . . . . . . . 14
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14 | 12, 13 | bitri 253 |
. . . . . . . . . . . . 13
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15 | 11, 14 | nemtbir 2718 |
. . . . . . . . . . . 12
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16 | nelneq2 2553 |
. . . . . . . . . . . 12
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17 | 9, 15, 16 | mp2an 677 |
. . . . . . . . . . 11
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18 | eqcom 2457 |
. . . . . . . . . . 11
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19 | 17, 18 | mtbi 300 |
. . . . . . . . . 10
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20 | biorf 407 |
. . . . . . . . . 10
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21 | 19, 20 | ax-mp 5 |
. . . . . . . . 9
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22 | 7, 21 | sylibr 216 |
. . . . . . . 8
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23 | 22 | preq2d 4057 |
. . . . . . 7
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24 | 1, 23 | eqtr4d 2487 |
. . . . . 6
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25 | prex 4641 |
. . . . . . 7
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26 | prex 4641 |
. . . . . . 7
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27 | 25, 26 | preqr1 4147 |
. . . . . 6
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28 | 24, 27 | syl 17 |
. . . . 5
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29 | snex 4640 |
. . . . . 6
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30 | snex 4640 |
. . . . . 6
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31 | 29, 30 | preqr1 4147 |
. . . . 5
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32 | 28, 31 | syl 17 |
. . . 4
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33 | opthw.1 |
. . . . 5
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34 | 33 | sneqr 4138 |
. . . 4
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35 | 32, 34 | syl 17 |
. . 3
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36 | snex 4640 |
. . . . . 6
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37 | 36 | sneqr 4138 |
. . . . 5
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38 | 22, 37 | syl 17 |
. . . 4
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39 | 10 | sneqr 4138 |
. . . 4
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40 | 38, 39 | syl 17 |
. . 3
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41 | 35, 40 | jca 535 |
. 2
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
42 | sneq 3977 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
43 | 42 | preq1d 4056 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
44 | 43 | preq1d 4056 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
45 | sneq 3977 |
. . . . 5
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46 | sneq 3977 |
. . . . 5
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47 | 45, 46 | syl 17 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
48 | 47 | preq2d 4057 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
49 | 44, 48 | sylan9eq 2504 |
. 2
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
50 | 41, 49 | impbii 191 |
1
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Colors of variables: wff setvar class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1668 ax-4 1681 ax-5 1757 ax-6 1804 ax-7 1850 ax-9 1895 ax-10 1914 ax-11 1919 ax-12 1932 ax-13 2090 ax-ext 2430 ax-sep 4524 ax-nul 4533 ax-pr 4638 |
This theorem depends on definitions: df-bi 189 df-or 372 df-an 373 df-tru 1446 df-ex 1663 df-nf 1667 df-sb 1797 df-clab 2437 df-cleq 2443 df-clel 2446 df-nfc 2580 df-ne 2623 df-v 3046 df-dif 3406 df-un 3408 df-nul 3731 df-sn 3968 df-pr 3970 |
This theorem is referenced by: (None) |
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