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Mirrors > Home > ILE Home > Th. List > iserd | Unicode version |
Description: A reflexive, symmetric, transitive relation is an equivalence relation on its domain. (Contributed by Mario Carneiro, 9-Jul-2014.) (Revised by Mario Carneiro, 12-Aug-2015.) |
Ref | Expression |
---|---|
iserd.1 |
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iserd.2 |
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iserd.3 |
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iserd.4 |
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Ref | Expression |
---|---|
iserd |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | iserd.1 |
. . 3
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2 | eqidd 2141 |
. . 3
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3 | iserd.2 |
. . . . . . . 8
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4 | 3 | ex 114 |
. . . . . . 7
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5 | iserd.3 |
. . . . . . . 8
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6 | 5 | ex 114 |
. . . . . . 7
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7 | 4, 6 | jca 304 |
. . . . . 6
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8 | 7 | alrimiv 1847 |
. . . . 5
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9 | 8 | alrimiv 1847 |
. . . 4
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10 | 9 | alrimiv 1847 |
. . 3
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11 | dfer2 6438 |
. . 3
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12 | 1, 2, 10, 11 | syl3anbrc 1166 |
. 2
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13 | 12 | adantr 274 |
. . . . . . . 8
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14 | simpr 109 |
. . . . . . . 8
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15 | 13, 14 | erref 6457 |
. . . . . . 7
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16 | 15 | ex 114 |
. . . . . 6
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17 | vex 2692 |
. . . . . . 7
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18 | 17, 17 | breldm 4751 |
. . . . . 6
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19 | 16, 18 | impbid1 141 |
. . . . 5
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20 | iserd.4 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
21 | 19, 20 | bitr4d 190 |
. . . 4
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22 | 21 | eqrdv 2138 |
. . 3
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23 | ereq2 6445 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
24 | 22, 23 | syl 14 |
. 2
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25 | 12, 24 | mpbid 146 |
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 105 ax-ia2 106 ax-ia3 107 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-sep 4054 ax-pow 4106 ax-pr 4139 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ral 2422 df-rex 2423 df-v 2691 df-un 3080 df-in 3082 df-ss 3089 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-br 3938 df-opab 3998 df-xp 4553 df-rel 4554 df-cnv 4555 df-co 4556 df-dm 4557 df-er 6437 |
This theorem is referenced by: swoer 6465 eqer 6469 0er 6471 iinerm 6509 erinxp 6511 ecopover 6535 ecopoverg 6538 ener 6681 enq0er 7267 xmeter 12644 |
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