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Mirrors > Home > MPE Home > Th. List > opth | Structured version Visualization version 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 4678 |
. . 3
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4 | 1, 2 | opi1 4672 |
. . . . . . 7
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5 | id 22 |
. . . . . . 7
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6 | 4, 5 | syl5eleq 2537 |
. . . . . 6
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7 | oprcl 4194 |
. . . . . 6
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8 | 6, 7 | syl 17 |
. . . . 5
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9 | 8 | simprd 465 |
. . . 4
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10 | 3 | opeq1d 4175 |
. . . . . . . 8
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11 | 10, 5 | eqtr3d 2489 |
. . . . . . 7
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12 | 8 | simpld 461 |
. . . . . . . 8
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13 | dfopg 4167 |
. . . . . . . 8
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14 | 12, 2, 13 | sylancl 669 |
. . . . . . 7
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15 | 11, 14 | eqtr3d 2489 |
. . . . . 6
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16 | dfopg 4167 |
. . . . . . 7
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17 | 8, 16 | syl 17 |
. . . . . 6
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18 | 15, 17 | eqtr3d 2489 |
. . . . 5
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19 | prex 4645 |
. . . . . 6
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20 | prex 4645 |
. . . . . 6
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21 | 19, 20 | preqr2 4153 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
22 | 18, 21 | syl 17 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
23 | preq2 4055 |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
24 | 23 | eqeq2d 2463 |
. . . . . 6
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25 | eqeq2 2464 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
26 | 24, 25 | imbi12d 322 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
27 | vex 3050 |
. . . . . 6
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28 | 2, 27 | preqr2 4153 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
29 | 26, 28 | vtoclg 3109 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
30 | 9, 22, 29 | sylc 62 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
31 | 3, 30 | jca 535 |
. 2
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
32 | opeq12 4171 |
. 2
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
33 | 31, 32 | impbii 191 |
1
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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 1671 ax-4 1684 ax-5 1760 ax-6 1807 ax-7 1853 ax-9 1898 ax-10 1917 ax-11 1922 ax-12 1935 ax-13 2093 ax-ext 2433 ax-sep 4528 ax-nul 4537 ax-pr 4642 |
This theorem depends on definitions: df-bi 189 df-or 372 df-an 373 df-3an 988 df-tru 1449 df-ex 1666 df-nf 1670 df-sb 1800 df-clab 2440 df-cleq 2446 df-clel 2449 df-nfc 2583 df-ne 2626 df-rab 2748 df-v 3049 df-dif 3409 df-un 3411 df-in 3413 df-ss 3420 df-nul 3734 df-if 3884 df-sn 3971 df-pr 3973 df-op 3977 |
This theorem is referenced by: opthg 4680 otth2 4686 copsexg 4690 copsex4g 4693 opcom 4698 moop2 4699 opelopabsbALT 4713 ralxpf 4984 cnvcnvsn 5316 funopg 5617 tpres 6122 oprabv 6344 xpopth 6837 eqop 6838 opiota 6857 soxp 6914 fnwelem 6916 xpdom2 7672 xpf1o 7739 unxpdomlem2 7782 unxpdomlem3 7783 xpwdomg 8105 fseqenlem1 8460 iundom2g 8970 eqresr 9566 cnref1o 11304 hashfun 12616 fsumcom2 13847 fprodcom2 14050 qredeu 14676 qnumdenbi 14705 crt 14738 prmreclem3 14874 imasaddfnlem 15446 dprd2da 17687 dprd2d2 17689 ucnima 21308 numclwlk1lem2f1 25834 br8d 28230 xppreima2 28261 aciunf1lem 28276 ofpreima 28280 erdszelem9 29934 msubff1 30206 mvhf1 30209 brtp 30401 br8 30408 br6 30409 br4 30410 brsegle 30887 poimirlem4 31956 poimirlem9 31961 f1opr 32063 dib1dim 34745 diclspsn 34774 dihopelvalcpre 34828 dihmeetlem4preN 34886 dihmeetlem13N 34899 dih1dimatlem 34909 dihatlat 34914 pellexlem3 35687 pellex 35691 snhesn 36394 opelopab4 36929 propssopi 39010 funsndifnop 39032 |
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