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Mirrors > Home > ILE Home > Th. List > ersym | Unicode version |
Description: An equivalence relation is symmetric. (Contributed by NM, 4-Jun-1995.) (Revised by Mario Carneiro, 12-Aug-2015.) |
Ref | Expression |
---|---|
ersym.1 |
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ersym.2 |
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Ref | Expression |
---|---|
ersym |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ersym.2 |
. . 3
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2 | ersym.1 |
. . . . . 6
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3 | errel 6315 |
. . . . . 6
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4 | 2, 3 | syl 14 |
. . . . 5
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5 | brrelex12 4489 |
. . . . 5
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6 | 4, 1, 5 | syl2anc 404 |
. . . 4
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7 | brcnvg 4630 |
. . . . 5
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8 | 7 | ancoms 265 |
. . . 4
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9 | 6, 8 | syl 14 |
. . 3
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10 | 1, 9 | mpbird 166 |
. 2
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11 | df-er 6306 |
. . . . . 6
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12 | 11 | simp3bi 961 |
. . . . 5
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13 | 2, 12 | syl 14 |
. . . 4
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14 | 13 | unssad 3178 |
. . 3
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15 | 14 | ssbrd 3892 |
. 2
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16 | 10, 15 | mpd 13 |
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 105 ax-ia2 106 ax-ia3 107 ax-io 666 ax-5 1382 ax-7 1383 ax-gen 1384 ax-ie1 1428 ax-ie2 1429 ax-8 1441 ax-10 1442 ax-11 1443 ax-i12 1444 ax-bndl 1445 ax-4 1446 ax-14 1451 ax-17 1465 ax-i9 1469 ax-ial 1473 ax-i5r 1474 ax-ext 2071 ax-sep 3963 ax-pow 4015 ax-pr 4045 |
This theorem depends on definitions: df-bi 116 df-3an 927 df-tru 1293 df-nf 1396 df-sb 1694 df-eu 1952 df-mo 1953 df-clab 2076 df-cleq 2082 df-clel 2085 df-nfc 2218 df-ral 2365 df-rex 2366 df-v 2622 df-un 3004 df-in 3006 df-ss 3013 df-pw 3435 df-sn 3456 df-pr 3457 df-op 3459 df-br 3852 df-opab 3906 df-xp 4458 df-rel 4459 df-cnv 4460 df-er 6306 |
This theorem is referenced by: ercl2 6319 ersymb 6320 ertr2d 6323 ertr3d 6324 ertr4d 6325 erth 6350 erinxp 6380 |
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