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Mirrors > Home > ILE Home > Th. List > ercnv | Unicode version |
Description: The converse of an equivalence relation is itself. (Contributed by Mario Carneiro, 12-Aug-2015.) |
Ref | Expression |
---|---|
ercnv |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | errel 6558 |
. 2
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2 | relcnv 5018 |
. . 3
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3 | id 19 |
. . . . . 6
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4 | 3 | ersymb 6563 |
. . . . 5
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5 | vex 2752 |
. . . . . . 7
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6 | vex 2752 |
. . . . . . 7
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7 | 5, 6 | brcnv 4822 |
. . . . . 6
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8 | df-br 4016 |
. . . . . 6
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9 | 7, 8 | bitr3i 186 |
. . . . 5
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10 | df-br 4016 |
. . . . 5
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11 | 4, 9, 10 | 3bitr3g 222 |
. . . 4
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12 | 11 | eqrelrdv2 4737 |
. . 3
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13 | 2, 12 | mpanl1 434 |
. 2
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14 | 1, 13 | mpancom 422 |
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 710 ax-5 1457 ax-7 1458 ax-gen 1459 ax-ie1 1503 ax-ie2 1504 ax-8 1514 ax-10 1515 ax-11 1516 ax-i12 1517 ax-bndl 1519 ax-4 1520 ax-17 1536 ax-i9 1540 ax-ial 1544 ax-i5r 1545 ax-14 2161 ax-ext 2169 ax-sep 4133 ax-pow 4186 ax-pr 4221 |
This theorem depends on definitions: df-bi 117 df-3an 981 df-tru 1366 df-nf 1471 df-sb 1773 df-eu 2039 df-mo 2040 df-clab 2174 df-cleq 2180 df-clel 2183 df-nfc 2318 df-ral 2470 df-rex 2471 df-v 2751 df-un 3145 df-in 3147 df-ss 3154 df-pw 3589 df-sn 3610 df-pr 3611 df-op 3613 df-br 4016 df-opab 4077 df-xp 4644 df-rel 4645 df-cnv 4646 df-er 6549 |
This theorem is referenced by: errn 6571 |
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