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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 4769 |
. . . 4
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2 | 1 | a1i 9 |
. . 3
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3 | df-br 4019 |
. . . . 5
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4 | noel 3441 |
. . . . . 6
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5 | 4 | pm2.21i 647 |
. . . . 5
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6 | 3, 5 | sylbi 121 |
. . . 4
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7 | 6 | adantl 277 |
. . 3
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8 | 4 | pm2.21i 647 |
. . . . 5
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9 | 3, 8 | sylbi 121 |
. . . 4
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10 | 9 | ad2antrl 490 |
. . 3
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11 | noel 3441 |
. . . . . 6
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12 | noel 3441 |
. . . . . 6
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13 | 11, 12 | 2false 702 |
. . . . 5
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14 | df-br 4019 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
15 | 13, 14 | bitr4i 187 |
. . . 4
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16 | 15 | a1i 9 |
. . 3
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17 | 2, 7, 10, 16 | iserd 6586 |
. 2
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18 | 17 | mptru 1373 |
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-in1 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-14 2163 ax-ext 2171 ax-sep 4136 ax-pow 4192 ax-pr 4227 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ral 2473 df-rex 2474 df-v 2754 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-nul 3438 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-br 4019 df-opab 4080 df-xp 4650 df-rel 4651 df-cnv 4652 df-co 4653 df-dm 4654 df-er 6560 |
This theorem is referenced by: (None) |
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