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Mirrors > Home > ILE Home > Th. List > intasym | Unicode version |
Description: Two ways of saying a relation is antisymmetric. Definition of antisymmetry in [Schechter] p. 51. (Contributed by NM, 9-Sep-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) |
Ref | Expression |
---|---|
intasym |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | relcnv 5027 |
. . 3
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2 | relin2 4766 |
. . 3
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3 | ssrel 4735 |
. . 3
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4 | 1, 2, 3 | mp2b 8 |
. 2
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5 | elin 3333 |
. . . . 5
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6 | df-br 4022 |
. . . . . 6
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7 | vex 2755 |
. . . . . . . 8
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8 | vex 2755 |
. . . . . . . 8
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9 | 7, 8 | brcnv 4831 |
. . . . . . 7
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10 | df-br 4022 |
. . . . . . 7
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11 | 9, 10 | bitr3i 186 |
. . . . . 6
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12 | 6, 11 | anbi12i 460 |
. . . . 5
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13 | 5, 12 | bitr4i 187 |
. . . 4
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14 | df-br 4022 |
. . . . 5
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15 | 8 | ideq 4800 |
. . . . 5
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16 | 14, 15 | bitr3i 186 |
. . . 4
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17 | 13, 16 | imbi12i 239 |
. . 3
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18 | 17 | 2albii 1482 |
. 2
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19 | 4, 18 | bitri 184 |
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 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 4139 ax-pow 4195 ax-pr 4230 |
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-un 3148 df-in 3150 df-ss 3157 df-pw 3595 df-sn 3616 df-pr 3617 df-op 3619 df-br 4022 df-opab 4083 df-id 4314 df-xp 4653 df-rel 4654 df-cnv 4655 |
This theorem is referenced by: (None) |
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