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Mirrors > Home > MPE Home > Th. List > opth1 | Structured version Visualization version Unicode version |
Description: Equality of the first members of equal ordered pairs. (Contributed by NM, 28-May-2008.) (Revised by Mario Carneiro, 26-Apr-2015.) |
Ref | Expression |
---|---|
opth1.1 |
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opth1.2 |
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Ref | Expression |
---|---|
opth1 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | opth1.1 |
. . . 4
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2 | 1 | sneqr 4139 |
. . 3
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3 | 2 | a1i 11 |
. 2
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4 | opth1.2 |
. . . . . . . . 9
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5 | 1, 4 | opi1 4669 |
. . . . . . . 8
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6 | id 22 |
. . . . . . . 8
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7 | 5, 6 | syl5eleq 2535 |
. . . . . . 7
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8 | oprcl 4191 |
. . . . . . 7
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9 | 7, 8 | syl 17 |
. . . . . 6
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10 | 9 | simpld 461 |
. . . . 5
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11 | prid1g 4078 |
. . . . 5
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12 | 10, 11 | syl 17 |
. . . 4
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13 | eleq2 2518 |
. . . 4
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14 | 12, 13 | syl5ibrcom 226 |
. . 3
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15 | elsni 3993 |
. . . 4
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16 | 15 | eqcomd 2457 |
. . 3
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17 | 14, 16 | syl6 34 |
. 2
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18 | dfopg 4164 |
. . . . 5
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19 | 7, 8, 18 | 3syl 18 |
. . . 4
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20 | 7, 19 | eleqtrd 2531 |
. . 3
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21 | elpri 3985 |
. . 3
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22 | 20, 21 | syl 17 |
. 2
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23 | 3, 17, 22 | mpjaod 383 |
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 1669 ax-4 1682 ax-5 1758 ax-6 1805 ax-7 1851 ax-9 1896 ax-10 1915 ax-11 1920 ax-12 1933 ax-13 2091 ax-ext 2431 ax-sep 4525 ax-nul 4534 ax-pr 4639 |
This theorem depends on definitions: df-bi 189 df-or 372 df-an 373 df-3an 987 df-tru 1447 df-ex 1664 df-nf 1668 df-sb 1798 df-clab 2438 df-cleq 2444 df-clel 2447 df-nfc 2581 df-ne 2624 df-v 3047 df-dif 3407 df-un 3409 df-in 3411 df-ss 3418 df-nul 3732 df-if 3882 df-sn 3969 df-pr 3971 df-op 3975 |
This theorem is referenced by: opth 4676 dmsnopg 5307 funcnvsn 5627 oprabid 6317 seqomlem2 7168 unxpdomlem3 7778 dfac5lem4 8557 dcomex 8877 canthwelem 9075 uzrdgfni 12172 gsum2d2 17606 2trllemA 25280 2pthon 25332 2pthon3v 25334 constr3lem2 25374 poimirlem9 31949 |
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