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Mirrors > Home > ILE Home > Th. List > dom0 | Unicode version |
Description: A set dominated by the empty set is empty. (Contributed by NM, 22-Nov-2004.) |
Ref | Expression |
---|---|
dom0 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | brdomi 6794 |
. . 3
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2 | f1f 5451 |
. . . . . 6
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3 | f00 5437 |
. . . . . 6
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4 | 2, 3 | sylib 122 |
. . . . 5
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5 | 4 | simprd 114 |
. . . 4
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6 | 5 | adantl 277 |
. . 3
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7 | 1, 6 | exlimddv 1910 |
. 2
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8 | 0ex 4156 |
. . . 4
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9 | domrefg 6812 |
. . . 4
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10 | 8, 9 | ax-mp 5 |
. . 3
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11 | breq1 4032 |
. . 3
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12 | 10, 11 | mpbiri 168 |
. 2
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13 | 7, 12 | impbii 126 |
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-13 2166 ax-14 2167 ax-ext 2175 ax-sep 4147 ax-nul 4155 ax-pow 4203 ax-pr 4238 ax-un 4462 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2045 df-mo 2046 df-clab 2180 df-cleq 2186 df-clel 2189 df-nfc 2325 df-ral 2477 df-rex 2478 df-v 2762 df-dif 3155 df-un 3157 df-in 3159 df-ss 3166 df-nul 3447 df-pw 3603 df-sn 3624 df-pr 3625 df-op 3627 df-uni 3836 df-br 4030 df-opab 4091 df-id 4322 df-xp 4661 df-rel 4662 df-cnv 4663 df-co 4664 df-dm 4665 df-rn 4666 df-res 4667 df-ima 4668 df-fun 5248 df-fn 5249 df-f 5250 df-f1 5251 df-fo 5252 df-f1o 5253 df-en 6786 df-dom 6787 |
This theorem is referenced by: (None) |
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