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Mirrors > Home > ILE Home > Th. List > opeqpr | Unicode version |
Description: Equivalence for an ordered pair equal to an unordered pair. (Contributed by NM, 3-Jun-2008.) |
Ref | Expression |
---|---|
opeqpr.1 |
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opeqpr.2 |
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opeqpr.3 |
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opeqpr.4 |
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Ref | Expression |
---|---|
opeqpr |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqcom 2179 |
. 2
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2 | opeqpr.1 |
. . . 4
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3 | opeqpr.2 |
. . . 4
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4 | 2, 3 | dfop 3777 |
. . 3
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5 | 4 | eqeq2i 2188 |
. 2
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6 | opeqpr.3 |
. . 3
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7 | opeqpr.4 |
. . 3
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8 | 2 | snex 4185 |
. . 3
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9 | prexg 4211 |
. . . 4
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10 | 2, 3, 9 | mp2an 426 |
. . 3
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11 | 6, 7, 8, 10 | preq12b 3770 |
. 2
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12 | 1, 5, 11 | 3bitri 206 |
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-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-14 2151 ax-ext 2159 ax-sep 4121 ax-pow 4174 ax-pr 4209 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1461 df-sb 1763 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-v 2739 df-un 3133 df-in 3135 df-ss 3142 df-pw 3577 df-sn 3598 df-pr 3599 df-op 3601 |
This theorem is referenced by: relop 4777 |
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