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Mathbox for Scott Fenton |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > brtxp | Structured version Visualization version Unicode version |
Description: Characterize a trinary relationship over a tail Cartesian product. Together with txpss3v 30638, this completely defines membership in a tail cross. (Contributed by Scott Fenton, 31-Mar-2012.) |
Ref | Expression |
---|---|
brtxp.1 |
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brtxp.2 |
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brtxp.3 |
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Ref | Expression |
---|---|
brtxp |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-txp 30613 |
. . 3
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2 | 1 | breqi 4407 |
. 2
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3 | brin 4451 |
. 2
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4 | brtxp.1 |
. . . . 5
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5 | opex 4663 |
. . . . 5
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6 | 4, 5 | brco 5004 |
. . . 4
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7 | ancom 452 |
. . . . . 6
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8 | vex 3047 |
. . . . . . . . 9
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9 | 8, 5 | brcnv 5016 |
. . . . . . . 8
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10 | brtxp.2 |
. . . . . . . . . 10
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11 | brtxp.3 |
. . . . . . . . . 10
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12 | 10, 11 | opelvv 4880 |
. . . . . . . . 9
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13 | 8 | brres 5110 |
. . . . . . . . 9
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14 | 12, 13 | mpbiran2 929 |
. . . . . . . 8
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15 | 10, 11, 8 | br1steq 30407 |
. . . . . . . 8
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16 | 9, 14, 15 | 3bitri 275 |
. . . . . . 7
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17 | 16 | anbi1i 700 |
. . . . . 6
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18 | 7, 17 | bitri 253 |
. . . . 5
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19 | 18 | exbii 1717 |
. . . 4
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20 | breq2 4405 |
. . . . 5
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21 | 10, 20 | ceqsexv 3083 |
. . . 4
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22 | 6, 19, 21 | 3bitri 275 |
. . 3
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23 | 4, 5 | brco 5004 |
. . . 4
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24 | ancom 452 |
. . . . . 6
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25 | vex 3047 |
. . . . . . . . 9
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26 | 25, 5 | brcnv 5016 |
. . . . . . . 8
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27 | 25 | brres 5110 |
. . . . . . . . 9
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28 | 12, 27 | mpbiran2 929 |
. . . . . . . 8
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29 | 10, 11, 25 | br2ndeq 30408 |
. . . . . . . 8
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30 | 26, 28, 29 | 3bitri 275 |
. . . . . . 7
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31 | 30 | anbi1i 700 |
. . . . . 6
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32 | 24, 31 | bitri 253 |
. . . . 5
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33 | 32 | exbii 1717 |
. . . 4
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34 | breq2 4405 |
. . . . 5
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35 | 11, 34 | ceqsexv 3083 |
. . . 4
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36 | 23, 33, 35 | 3bitri 275 |
. . 3
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37 | 22, 36 | anbi12i 702 |
. 2
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38 | 2, 3, 37 | 3bitri 275 |
1
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Colors of variables: wff setvar class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1668 ax-4 1681 ax-5 1757 ax-6 1804 ax-7 1850 ax-8 1888 ax-9 1895 ax-10 1914 ax-11 1919 ax-12 1932 ax-13 2090 ax-ext 2430 ax-sep 4524 ax-nul 4533 ax-pow 4580 ax-pr 4638 ax-un 6580 |
This theorem depends on definitions: df-bi 189 df-or 372 df-an 373 df-3an 986 df-tru 1446 df-ex 1663 df-nf 1667 df-sb 1797 df-eu 2302 df-mo 2303 df-clab 2437 df-cleq 2443 df-clel 2446 df-nfc 2580 df-ne 2623 df-ral 2741 df-rex 2742 df-rab 2745 df-v 3046 df-sbc 3267 df-dif 3406 df-un 3408 df-in 3410 df-ss 3417 df-nul 3731 df-if 3881 df-sn 3968 df-pr 3970 df-op 3974 df-uni 4198 df-br 4402 df-opab 4461 df-mpt 4462 df-id 4748 df-xp 4839 df-rel 4840 df-cnv 4841 df-co 4842 df-dm 4843 df-rn 4844 df-res 4845 df-iota 5545 df-fun 5583 df-fn 5584 df-f 5585 df-fo 5587 df-fv 5589 df-1st 6790 df-2nd 6791 df-txp 30613 |
This theorem is referenced by: brtxp2 30641 pprodss4v 30644 brpprod 30645 brsset 30649 brtxpsd 30654 elfuns 30675 |
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