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Mirrors > Home > ILE Home > Th. List > opth | Unicode version |
Description: The ordered pair theorem.
If two ordered pairs are equal, their first
elements are equal and their second elements are equal. Exercise 6 of
[TakeutiZaring] p. 16. Note that
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Ref | Expression |
---|---|
opth1.1 |
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opth1.2 |
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Ref | Expression |
---|---|
opth |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | opth1.1 |
. . . 4
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2 | opth1.2 |
. . . 4
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3 | 1, 2 | opth1 4166 |
. . 3
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4 | 1, 2 | opi1 4162 |
. . . . . . 7
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5 | id 19 |
. . . . . . 7
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6 | 4, 5 | eleqtrid 2229 |
. . . . . 6
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7 | oprcl 3737 |
. . . . . 6
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8 | 6, 7 | syl 14 |
. . . . 5
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9 | 8 | simprd 113 |
. . . 4
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10 | 3 | opeq1d 3719 |
. . . . . . . 8
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11 | 10, 5 | eqtr3d 2175 |
. . . . . . 7
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12 | 8 | simpld 111 |
. . . . . . . 8
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13 | dfopg 3711 |
. . . . . . . 8
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14 | 12, 2, 13 | sylancl 410 |
. . . . . . 7
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15 | 11, 14 | eqtr3d 2175 |
. . . . . 6
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16 | dfopg 3711 |
. . . . . . 7
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17 | 8, 16 | syl 14 |
. . . . . 6
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18 | 15, 17 | eqtr3d 2175 |
. . . . 5
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19 | prexg 4141 |
. . . . . . 7
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20 | 12, 2, 19 | sylancl 410 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
21 | prexg 4141 |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
22 | 8, 21 | syl 14 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
23 | preqr2g 3702 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
24 | 20, 22, 23 | syl2anc 409 |
. . . . 5
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25 | 18, 24 | mpd 13 |
. . . 4
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26 | preq2 3609 |
. . . . . . 7
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27 | 26 | eqeq2d 2152 |
. . . . . 6
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28 | eqeq2 2150 |
. . . . . 6
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29 | 27, 28 | imbi12d 233 |
. . . . 5
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30 | vex 2692 |
. . . . . 6
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31 | 2, 30 | preqr2 3704 |
. . . . 5
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32 | 29, 31 | vtoclg 2749 |
. . . 4
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33 | 9, 25, 32 | sylc 62 |
. . 3
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34 | 3, 33 | jca 304 |
. 2
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35 | opeq12 3715 |
. 2
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36 | 34, 35 | impbii 125 |
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 105 ax-ia2 106 ax-ia3 107 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-sep 4054 ax-pow 4106 ax-pr 4139 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-nf 1438 df-sb 1737 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-v 2691 df-un 3080 df-in 3082 df-ss 3089 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 |
This theorem is referenced by: opthg 4168 otth2 4171 copsexg 4174 copsex4g 4177 opcom 4180 moop2 4181 opelopabsbALT 4189 opelopabsb 4190 ralxpf 4693 rexxpf 4694 cnvcnvsn 5023 funopg 5165 funinsn 5180 brabvv 5825 xpdom2 6733 xpf1o 6746 djuf1olem 6946 enq0ref 7265 enq0tr 7266 mulnnnq0 7282 eqresr 7668 cnref1o 9469 fisumcom2 11239 qredeu 11814 |
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