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Mirrors > Home > MPE Home > Th. List > brdom3 | Structured version Visualization version Unicode version |
Description: Equivalence to a dominance relation. (Contributed by NM, 27-Mar-2007.) |
Ref | Expression |
---|---|
brdom3.2 |
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Ref | Expression |
---|---|
brdom3 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | reldom 7575 |
. . . . . . . . 9
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2 | 1 | brrelexi 4875 |
. . . . . . . 8
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3 | 0sdomg 7701 |
. . . . . . . 8
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4 | 2, 3 | syl 17 |
. . . . . . 7
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5 | df-ne 2624 |
. . . . . . 7
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6 | 4, 5 | syl6bb 265 |
. . . . . 6
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7 | 6 | biimpar 488 |
. . . . 5
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8 | fodomr 7723 |
. . . . . 6
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9 | 8 | ancoms 455 |
. . . . 5
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10 | 7, 9 | syldan 473 |
. . . 4
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11 | pm5.6 923 |
. . . 4
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12 | 10, 11 | mpbi 212 |
. . 3
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13 | br0 4449 |
. . . . . . . 8
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14 | 13 | nex 1678 |
. . . . . . 7
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15 | exmo 2324 |
. . . . . . 7
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16 | 14, 15 | mtpor 1653 |
. . . . . 6
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17 | 16 | ax-gen 1669 |
. . . . 5
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18 | rzal 3871 |
. . . . 5
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19 | 0ex 4535 |
. . . . . 6
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20 | breq 4404 |
. . . . . . . . 9
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21 | 20 | mobidv 2320 |
. . . . . . . 8
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22 | 21 | albidv 1767 |
. . . . . . 7
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23 | breq 4404 |
. . . . . . . . 9
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24 | 23 | rexbidv 2901 |
. . . . . . . 8
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25 | 24 | ralbidv 2827 |
. . . . . . 7
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26 | 22, 25 | anbi12d 717 |
. . . . . 6
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27 | 19, 26 | spcev 3141 |
. . . . 5
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28 | 17, 18, 27 | sylancr 669 |
. . . 4
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29 | fofun 5794 |
. . . . . . 7
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30 | dffun6 5597 |
. . . . . . . 8
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31 | 30 | simprbi 466 |
. . . . . . 7
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32 | 29, 31 | syl 17 |
. . . . . 6
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33 | dffo4 6038 |
. . . . . . 7
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34 | 33 | simprbi 466 |
. . . . . 6
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35 | 32, 34 | jca 535 |
. . . . 5
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36 | 35 | eximi 1707 |
. . . 4
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37 | 28, 36 | jaoi 381 |
. . 3
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38 | 12, 37 | syl 17 |
. 2
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39 | inss1 3652 |
. . . . . . . . . . 11
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40 | 39 | ssbri 4445 |
. . . . . . . . . 10
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41 | 40 | moimi 2349 |
. . . . . . . . 9
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42 | 41 | alimi 1684 |
. . . . . . . 8
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43 | relxp 4942 |
. . . . . . . . . 10
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44 | relin2 4952 |
. . . . . . . . . 10
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45 | 43, 44 | ax-mp 5 |
. . . . . . . . 9
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46 | dffun6 5597 |
. . . . . . . . 9
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47 | 45, 46 | mpbiran 929 |
. . . . . . . 8
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48 | 42, 47 | sylibr 216 |
. . . . . . 7
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49 | funfn 5611 |
. . . . . . 7
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50 | 48, 49 | sylib 200 |
. . . . . 6
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51 | rninxp 5276 |
. . . . . . 7
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52 | 51 | biimpri 210 |
. . . . . 6
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53 | 50, 52 | anim12i 570 |
. . . . 5
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54 | df-fo 5588 |
. . . . 5
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55 | 53, 54 | sylibr 216 |
. . . 4
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56 | vex 3048 |
. . . . . . 7
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57 | 56 | inex1 4544 |
. . . . . 6
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58 | 57 | dmex 6726 |
. . . . 5
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59 | 58 | fodom 8952 |
. . . 4
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60 | brdom3.2 |
. . . . . 6
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61 | inss2 3653 |
. . . . . . . 8
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62 | dmss 5034 |
. . . . . . . 8
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63 | 61, 62 | ax-mp 5 |
. . . . . . 7
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64 | dmxpss 5268 |
. . . . . . 7
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65 | 63, 64 | sstri 3441 |
. . . . . 6
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66 | ssdomg 7615 |
. . . . . 6
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67 | 60, 65, 66 | mp2 9 |
. . . . 5
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68 | domtr 7622 |
. . . . 5
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69 | 67, 68 | mpan2 677 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
70 | 55, 59, 69 | 3syl 18 |
. . 3
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71 | 70 | exlimiv 1776 |
. 2
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72 | 38, 71 | impbii 191 |
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 1669 ax-4 1682 ax-5 1758 ax-6 1805 ax-7 1851 ax-8 1889 ax-9 1896 ax-10 1915 ax-11 1920 ax-12 1933 ax-13 2091 ax-ext 2431 ax-rep 4515 ax-sep 4525 ax-nul 4534 ax-pow 4581 ax-pr 4639 ax-un 6583 ax-ac2 8893 |
This theorem depends on definitions: df-bi 189 df-or 372 df-an 373 df-3or 986 df-3an 987 df-tru 1447 df-ex 1664 df-nf 1668 df-sb 1798 df-eu 2303 df-mo 2304 df-clab 2438 df-cleq 2444 df-clel 2447 df-nfc 2581 df-ne 2624 df-ral 2742 df-rex 2743 df-reu 2744 df-rmo 2745 df-rab 2746 df-v 3047 df-sbc 3268 df-csb 3364 df-dif 3407 df-un 3409 df-in 3411 df-ss 3418 df-pss 3420 df-nul 3732 df-if 3882 df-pw 3953 df-sn 3969 df-pr 3971 df-tp 3973 df-op 3975 df-uni 4199 df-int 4235 df-iun 4280 df-br 4403 df-opab 4462 df-mpt 4463 df-tr 4498 df-eprel 4745 df-id 4749 df-po 4755 df-so 4756 df-fr 4793 df-se 4794 df-we 4795 df-xp 4840 df-rel 4841 df-cnv 4842 df-co 4843 df-dm 4844 df-rn 4845 df-res 4846 df-ima 4847 df-pred 5380 df-ord 5426 df-on 5427 df-suc 5429 df-iota 5546 df-fun 5584 df-fn 5585 df-f 5586 df-f1 5587 df-fo 5588 df-f1o 5589 df-fv 5590 df-isom 5591 df-riota 6252 df-ov 6293 df-oprab 6294 df-mpt2 6295 df-1st 6793 df-2nd 6794 df-wrecs 7028 df-recs 7090 df-er 7363 df-map 7474 df-en 7570 df-dom 7571 df-sdom 7572 df-card 8373 df-acn 8376 df-ac 8547 |
This theorem is referenced by: brdom5 8957 brdom4 8958 |
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