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Mirrors > Home > ILE Home > Th. List > cnvsng | Unicode version |
Description: Converse of a singleton of an ordered pair. (Contributed by NM, 23-Jan-2015.) |
Ref | Expression |
---|---|
cnvsng |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | opeq1 3599 |
. . . . 5
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2 | 1 | sneqd 3438 |
. . . 4
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3 | 2 | cnveqd 4573 |
. . 3
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4 | opeq2 3600 |
. . . 4
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5 | 4 | sneqd 3438 |
. . 3
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6 | 3, 5 | eqeq12d 2099 |
. 2
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7 | opeq2 3600 |
. . . . 5
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8 | 7 | sneqd 3438 |
. . . 4
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9 | 8 | cnveqd 4573 |
. . 3
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10 | opeq1 3599 |
. . . 4
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11 | 10 | sneqd 3438 |
. . 3
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12 | 9, 11 | eqeq12d 2099 |
. 2
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13 | vex 2617 |
. . 3
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14 | vex 2617 |
. . 3
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15 | 13, 14 | cnvsn 4870 |
. 2
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16 | 6, 12, 15 | vtocl2g 2675 |
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 104 ax-ia2 105 ax-ia3 106 ax-io 663 ax-5 1379 ax-7 1380 ax-gen 1381 ax-ie1 1425 ax-ie2 1426 ax-8 1438 ax-10 1439 ax-11 1440 ax-i12 1441 ax-bndl 1442 ax-4 1443 ax-14 1448 ax-17 1462 ax-i9 1466 ax-ial 1470 ax-i5r 1471 ax-ext 2067 ax-sep 3925 ax-pow 3977 ax-pr 4003 |
This theorem depends on definitions: df-bi 115 df-3an 924 df-tru 1290 df-nf 1393 df-sb 1690 df-eu 1948 df-mo 1949 df-clab 2072 df-cleq 2078 df-clel 2081 df-nfc 2214 df-ral 2360 df-rex 2361 df-v 2616 df-un 2990 df-in 2992 df-ss 2999 df-pw 3411 df-sn 3431 df-pr 3432 df-op 3434 df-br 3815 df-opab 3869 df-xp 4410 df-rel 4411 df-cnv 4412 |
This theorem is referenced by: opswapg 4874 funsng 5016 |
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