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Mirrors > Home > ILE Home > Th. List > 0er | Unicode version |
Description: The empty set is an equivalence relation on the empty set. (Contributed by Mario Carneiro, 5-Sep-2015.) |
Ref | Expression |
---|---|
0er |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rel0 4592 |
. . . 4
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2 | 1 | a1i 9 |
. . 3
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3 | df-br 3868 |
. . . . 5
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4 | noel 3306 |
. . . . . 6
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5 | 4 | pm2.21i 613 |
. . . . 5
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6 | 3, 5 | sylbi 120 |
. . . 4
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7 | 6 | adantl 272 |
. . 3
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8 | 4 | pm2.21i 613 |
. . . . 5
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9 | 3, 8 | sylbi 120 |
. . . 4
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10 | 9 | ad2antrl 475 |
. . 3
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11 | noel 3306 |
. . . . . 6
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12 | noel 3306 |
. . . . . 6
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13 | 11, 12 | 2false 655 |
. . . . 5
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14 | df-br 3868 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
15 | 13, 14 | bitr4i 186 |
. . . 4
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16 | 15 | a1i 9 |
. . 3
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17 | 2, 7, 10, 16 | iserd 6358 |
. 2
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18 | 17 | mptru 1305 |
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-in1 582 ax-in2 583 ax-io 668 ax-5 1388 ax-7 1389 ax-gen 1390 ax-ie1 1434 ax-ie2 1435 ax-8 1447 ax-10 1448 ax-11 1449 ax-i12 1450 ax-bndl 1451 ax-4 1452 ax-14 1457 ax-17 1471 ax-i9 1475 ax-ial 1479 ax-i5r 1480 ax-ext 2077 ax-sep 3978 ax-pow 4030 ax-pr 4060 |
This theorem depends on definitions: df-bi 116 df-3an 929 df-tru 1299 df-nf 1402 df-sb 1700 df-eu 1958 df-mo 1959 df-clab 2082 df-cleq 2088 df-clel 2091 df-nfc 2224 df-ral 2375 df-rex 2376 df-v 2635 df-dif 3015 df-un 3017 df-in 3019 df-ss 3026 df-nul 3303 df-pw 3451 df-sn 3472 df-pr 3473 df-op 3475 df-br 3868 df-opab 3922 df-xp 4473 df-rel 4474 df-cnv 4475 df-co 4476 df-dm 4477 df-er 6332 |
This theorem is referenced by: (None) |
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