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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 3712 |
. . 3
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4 | 3 | eqeq1i 2148 |
. 2
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5 | 1 | snex 4117 |
. . 3
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6 | prexg 4141 |
. . . 4
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7 | 1, 2, 6 | mp2an 423 |
. . 3
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8 | opeqsn.3 |
. . 3
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9 | 5, 7, 8 | preqsn 3710 |
. 2
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10 | eqcom 2142 |
. . . . 5
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11 | 1, 2, 1 | preqsn 3710 |
. . . . 5
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12 | eqcom 2142 |
. . . . . . 7
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13 | 12 | anbi2i 453 |
. . . . . 6
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14 | anidm 394 |
. . . . . 6
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15 | 13, 14 | bitri 183 |
. . . . 5
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16 | 10, 11, 15 | 3bitri 205 |
. . . 4
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17 | 16 | anbi1i 454 |
. . 3
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18 | dfsn2 3546 |
. . . . . . 7
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19 | preq2 3609 |
. . . . . . 7
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20 | 18, 19 | syl5req 2186 |
. . . . . 6
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21 | 20 | eqeq1d 2149 |
. . . . 5
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22 | eqcom 2142 |
. . . . 5
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23 | 21, 22 | syl6bb 195 |
. . . 4
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24 | 23 | pm5.32i 450 |
. . 3
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25 | 17, 24 | bitri 183 |
. 2
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26 | 4, 9, 25 | 3bitri 205 |
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: relop 4697 |
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