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Mirrors > Home > ILE Home > Th. List > opeqsn | Unicode version |
Description: Equivalence for an ordered pair equal to a singleton. (Contributed by NM, 3-Jun-2008.) |
Ref | Expression |
---|---|
opeqsn.1 |
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opeqsn.2 |
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opeqsn.3 |
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Ref | Expression |
---|---|
opeqsn |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | opeqsn.1 |
. . . 4
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2 | opeqsn.2 |
. . . 4
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3 | 1, 2 | dfop 3539 |
. . 3
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4 | 3 | eqeq1i 2044 |
. 2
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5 | snexgOLD 3926 |
. . . 4
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6 | 1, 5 | ax-mp 7 |
. . 3
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7 | prexgOLD 3937 |
. . . 4
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8 | 1, 2, 7 | mp2an 402 |
. . 3
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9 | opeqsn.3 |
. . 3
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10 | 6, 8, 9 | preqsn 3537 |
. 2
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11 | eqcom 2039 |
. . . . 5
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12 | 1, 2, 1 | preqsn 3537 |
. . . . 5
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13 | eqcom 2039 |
. . . . . . 7
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14 | 13 | anbi2i 430 |
. . . . . 6
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15 | anidm 376 |
. . . . . 6
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16 | 14, 15 | bitri 173 |
. . . . 5
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17 | 11, 12, 16 | 3bitri 195 |
. . . 4
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18 | 17 | anbi1i 431 |
. . 3
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19 | dfsn2 3381 |
. . . . . . 7
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20 | preq2 3439 |
. . . . . . 7
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21 | 19, 20 | syl5req 2082 |
. . . . . 6
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22 | 21 | eqeq1d 2045 |
. . . . 5
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23 | eqcom 2039 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
24 | 22, 23 | syl6bb 185 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
25 | 24 | pm5.32i 427 |
. . 3
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26 | 18, 25 | bitri 173 |
. 2
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27 | 4, 10, 26 | 3bitri 195 |
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 ax-pr 3935 |
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: relop 4429 |
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