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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 4571 |
. . . . 5
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2 | eleq12 2254 |
. . . . . 6
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3 | 2 | anbi1d 465 |
. . . . 5
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4 | 1, 3 | mtbiri 676 |
. . . 4
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5 | 4 | con2i 628 |
. . 3
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6 | 5 | adantr 276 |
. 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 3785 |
. . . 4
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12 | 11 | biimpi 120 |
. . 3
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13 | 12 | adantl 277 |
. 2
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14 | 6, 13 | ecased 1360 |
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-in1 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-ext 2171 ax-setind 4554 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-nf 1472 df-sb 1774 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ral 2473 df-v 2754 df-dif 3146 df-un 3148 df-sn 3613 df-pr 3614 |
This theorem is referenced by: opthreg 4573 |
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