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Mirrors > Home > ILE Home > Th. List > domtr | Unicode version |
Description: Transitivity of dominance relation. Theorem 17 of [Suppes] p. 94. (Contributed by NM, 4-Jun-1998.) (Revised by Mario Carneiro, 15-Nov-2014.) |
Ref | Expression |
---|---|
domtr |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | reldom 6226 |
. 2
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2 | vex 2560 |
. . . 4
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3 | 2 | brdom 6231 |
. . 3
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4 | vex 2560 |
. . . 4
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5 | 4 | brdom 6231 |
. . 3
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6 | eeanv 1807 |
. . . 4
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7 | f1co 5101 |
. . . . . . . 8
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8 | 7 | ancoms 255 |
. . . . . . 7
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9 | vex 2560 |
. . . . . . . . 9
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10 | vex 2560 |
. . . . . . . . 9
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11 | 9, 10 | coex 4863 |
. . . . . . . 8
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12 | f1eq1 5087 |
. . . . . . . 8
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13 | 11, 12 | spcev 2647 |
. . . . . . 7
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14 | 8, 13 | syl 14 |
. . . . . 6
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15 | 4 | brdom 6231 |
. . . . . 6
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16 | 14, 15 | sylibr 137 |
. . . . 5
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17 | 16 | exlimivv 1776 |
. . . 4
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18 | 6, 17 | sylbir 125 |
. . 3
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19 | 3, 5, 18 | syl2anb 275 |
. 2
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20 | 1, 19 | vtoclr 4388 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-13 1404 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 ax-pow 3927 ax-pr 3944 ax-un 4170 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ral 2311 df-rex 2312 df-v 2559 df-un 2922 df-in 2924 df-ss 2931 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-uni 3581 df-br 3765 df-opab 3819 df-id 4030 df-xp 4351 df-rel 4352 df-cnv 4353 df-co 4354 df-dm 4355 df-rn 4356 df-fun 4904 df-fn 4905 df-f 4906 df-f1 4907 df-dom 6223 |
This theorem is referenced by: endomtr 6270 domentr 6271 nndomo 6326 |
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