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Mirrors > Home > ILE Home > Th. List > opth1 | 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 3522 |
. . 3
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3 | 2 | a1i 9 |
. 2
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4 | opth1.2 |
. . . . . . . . 9
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5 | 1, 4 | opi1 3960 |
. . . . . . . 8
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6 | id 19 |
. . . . . . . 8
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7 | 5, 6 | syl5eleq 2123 |
. . . . . . 7
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8 | oprcl 3564 |
. . . . . . 7
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9 | 7, 8 | syl 14 |
. . . . . 6
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10 | 9 | simpld 105 |
. . . . 5
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11 | prid1g 3465 |
. . . . 5
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12 | 10, 11 | syl 14 |
. . . 4
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13 | eleq2 2098 |
. . . 4
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14 | 12, 13 | syl5ibrcom 146 |
. . 3
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15 | elsni 3391 |
. . . 4
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16 | 15 | eqcomd 2042 |
. . 3
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17 | 14, 16 | syl6 29 |
. 2
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18 | dfopg 3538 |
. . . . 5
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19 | 7, 8, 18 | 3syl 17 |
. . . 4
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20 | 7, 19 | eleqtrd 2113 |
. . 3
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21 | elpri 3387 |
. . 3
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22 | 20, 21 | syl 14 |
. 2
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23 | 3, 17, 22 | mpjaod 637 |
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 99 ax-ia2 100 ax-ia3 101 ax-io 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-14 1402 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 ax-sep 3866 ax-pow 3918 |
This theorem depends on definitions: df-bi 110 df-3an 886 df-tru 1245 df-nf 1347 df-sb 1643 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-v 2553 df-un 2916 df-in 2918 df-ss 2925 df-pw 3353 df-sn 3373 df-pr 3374 df-op 3376 |
This theorem is referenced by: opth 3965 dmsnopg 4735 funcnvsn 4888 oprabid 5480 |
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