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Mirrors > Home > ILE Home > Th. List > intirr | Unicode version |
Description: Two ways of saying a relation is irreflexive. Definition of irreflexivity in [Schechter] p. 51. (Contributed by NM, 9-Sep-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) |
Ref | Expression |
---|---|
intirr |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | incom 3342 |
. . . 4
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2 | 1 | eqeq1i 2197 |
. . 3
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3 | disj2 3493 |
. . 3
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4 | reli 4771 |
. . . 4
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5 | ssrel 4729 |
. . . 4
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6 | 4, 5 | ax-mp 5 |
. . 3
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7 | 2, 3, 6 | 3bitri 206 |
. 2
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8 | equcom 1717 |
. . . . 5
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9 | vex 2755 |
. . . . . 6
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10 | 9 | ideq 4794 |
. . . . 5
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11 | df-br 4019 |
. . . . 5
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12 | 8, 10, 11 | 3bitr2i 208 |
. . . 4
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13 | vex 2755 |
. . . . . . . 8
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14 | 13, 9 | opex 4244 |
. . . . . . 7
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15 | 14 | biantrur 303 |
. . . . . 6
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16 | eldif 3153 |
. . . . . 6
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17 | 15, 16 | bitr4i 187 |
. . . . 5
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18 | df-br 4019 |
. . . . 5
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19 | 17, 18 | xchnxbir 682 |
. . . 4
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20 | 12, 19 | imbi12i 239 |
. . 3
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21 | 20 | 2albii 1482 |
. 2
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22 | nfv 1539 |
. . . 4
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23 | breq2 4022 |
. . . . 5
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24 | 23 | notbid 668 |
. . . 4
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25 | 22, 24 | equsal 1738 |
. . 3
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26 | 25 | albii 1481 |
. 2
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27 | 7, 21, 26 | 3bitr2i 208 |
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 4189 ax-pr 4224 |
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-id 4308 df-xp 4647 df-rel 4648 |
This theorem is referenced by: (None) |
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