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Mirrors > Home > ILE Home > Th. List > opthpr | Unicode version |
Description: A way to represent ordered pairs using unordered pairs with distinct members. (Contributed by NM, 27-Mar-2007.) |
Ref | Expression |
---|---|
preq12b.1 |
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preq12b.2 |
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preq12b.3 |
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preq12b.4 |
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Ref | Expression |
---|---|
opthpr |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | preq12b.1 |
. . 3
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2 | preq12b.2 |
. . 3
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3 | preq12b.3 |
. . 3
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4 | preq12b.4 |
. . 3
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5 | 1, 2, 3, 4 | preq12b 3768 |
. 2
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6 | idd 21 |
. . . 4
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7 | df-ne 2348 |
. . . . . 6
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8 | pm2.21 617 |
. . . . . 6
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9 | 7, 8 | sylbi 121 |
. . . . 5
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10 | 9 | impd 254 |
. . . 4
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11 | 6, 10 | jaod 717 |
. . 3
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12 | orc 712 |
. . 3
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13 | 11, 12 | impbid1 142 |
. 2
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14 | 5, 13 | bitrid 192 |
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 106 ax-ia2 107 ax-ia3 108 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-ext 2159 |
This theorem depends on definitions: df-bi 117 df-tru 1356 df-nf 1461 df-sb 1763 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-v 2739 df-un 3133 df-sn 3597 df-pr 3598 |
This theorem is referenced by: (None) |
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