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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 6738 |
. 2
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2 | vex 2740 |
. . . 4
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3 | 2 | brdom 6743 |
. . 3
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4 | vex 2740 |
. . . 4
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5 | 4 | brdom 6743 |
. . 3
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6 | eeanv 1932 |
. . . 4
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7 | f1co 5428 |
. . . . . . . 8
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8 | 7 | ancoms 268 |
. . . . . . 7
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9 | vex 2740 |
. . . . . . . . 9
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10 | vex 2740 |
. . . . . . . . 9
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11 | 9, 10 | coex 5169 |
. . . . . . . 8
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12 | f1eq1 5411 |
. . . . . . . 8
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13 | 11, 12 | spcev 2832 |
. . . . . . 7
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14 | 8, 13 | syl 14 |
. . . . . 6
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15 | 4 | brdom 6743 |
. . . . . 6
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16 | 14, 15 | sylibr 134 |
. . . . 5
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17 | 16 | exlimivv 1896 |
. . . 4
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18 | 6, 17 | sylbir 135 |
. . 3
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19 | 3, 5, 18 | syl2anb 291 |
. 2
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20 | 1, 19 | vtoclr 4670 |
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 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4118 ax-pow 4171 ax-pr 4205 ax-un 4429 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ral 2460 df-rex 2461 df-v 2739 df-un 3133 df-in 3135 df-ss 3142 df-pw 3576 df-sn 3597 df-pr 3598 df-op 3600 df-uni 3808 df-br 4001 df-opab 4062 df-id 4289 df-xp 4628 df-rel 4629 df-cnv 4630 df-co 4631 df-dm 4632 df-rn 4633 df-fun 5213 df-fn 5214 df-f 5215 df-f1 5216 df-dom 6735 |
This theorem is referenced by: endomtr 6783 domentr 6784 cnvct 6802 ssct 6811 nndomo 6857 infnfi 6888 xpct 12367 |
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