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Mirrors > Home > ILE Home > Th. List > preleq | Unicode version |
Description: Equality of two unordered pairs when one member of each pair contains the other member. (Contributed by NM, 16-Oct-1996.) |
Ref | Expression |
---|---|
preleq.1 |
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preleq.2 |
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preleq.3 |
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preleq.4 |
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Ref | Expression |
---|---|
preleq |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | en2lp 4477 |
. . . . 5
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2 | eleq12 2205 |
. . . . . 6
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3 | 2 | anbi1d 461 |
. . . . 5
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4 | 1, 3 | mtbiri 665 |
. . . 4
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5 | 4 | con2i 617 |
. . 3
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6 | 5 | adantr 274 |
. 2
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7 | preleq.1 |
. . . . 5
![]() ![]() ![]() ![]() | |
8 | preleq.2 |
. . . . 5
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9 | preleq.3 |
. . . . 5
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10 | preleq.4 |
. . . . 5
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11 | 7, 8, 9, 10 | preq12b 3705 |
. . . 4
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12 | 11 | biimpi 119 |
. . 3
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13 | 12 | adantl 275 |
. 2
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14 | 6, 13 | ecased 1328 |
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-in1 604 ax-in2 605 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-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-setind 4460 |
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-ral 2422 df-v 2691 df-dif 3078 df-un 3080 df-sn 3538 df-pr 3539 |
This theorem is referenced by: opthreg 4479 |
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