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Mirrors > Home > ILE Home > Th. List > preqsn | Unicode version |
Description: Equivalence for a pair equal to a singleton. (Contributed by NM, 3-Jun-2008.) |
Ref | Expression |
---|---|
preqsn.1 |
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preqsn.2 |
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preqsn.3 |
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Ref | Expression |
---|---|
preqsn |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfsn2 3381 |
. . 3
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2 | 1 | eqeq2i 2047 |
. 2
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3 | preqsn.1 |
. . . 4
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4 | preqsn.2 |
. . . 4
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5 | preqsn.3 |
. . . 4
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6 | 3, 4, 5, 5 | preq12b 3532 |
. . 3
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7 | oridm 673 |
. . . 4
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8 | eqtr3 2056 |
. . . . . 6
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9 | simpr 103 |
. . . . . 6
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10 | 8, 9 | jca 290 |
. . . . 5
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11 | eqtr 2054 |
. . . . . 6
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12 | simpr 103 |
. . . . . 6
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13 | 11, 12 | jca 290 |
. . . . 5
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14 | 10, 13 | impbii 117 |
. . . 4
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15 | 7, 14 | bitri 173 |
. . 3
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16 | 6, 15 | bitri 173 |
. 2
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17 | 2, 16 | bitri 173 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 |
This theorem depends on definitions: df-bi 110 df-tru 1245 df-nf 1347 df-sb 1643 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-v 2553 df-un 2916 df-sn 3373 df-pr 3374 |
This theorem is referenced by: opeqsn 3980 relop 4429 |
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