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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 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | brdomi 6724 | . . 3 | |
2 | f1f 5401 | . . . . . 6 | |
3 | f00 5387 | . . . . . 6 | |
4 | 2, 3 | sylib 121 | . . . . 5 |
5 | 4 | simprd 113 | . . . 4 |
6 | 5 | adantl 275 | . . 3 |
7 | 1, 6 | exlimddv 1891 | . 2 |
8 | 0ex 4114 | . . . 4 | |
9 | domrefg 6742 | . . . 4 | |
10 | 8, 9 | ax-mp 5 | . . 3 |
11 | breq1 3990 | . . 3 | |
12 | 10, 11 | mpbiri 167 | . 2 |
13 | 7, 12 | impbii 125 | 1 |
Colors of variables: wff set class |
Syntax hints: wa 103 wb 104 wceq 1348 wcel 2141 cvv 2730 c0 3414 class class class wbr 3987 wf 5192 wf1 5193 cdom 6714 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-sep 4105 ax-nul 4113 ax-pow 4158 ax-pr 4192 ax-un 4416 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ral 2453 df-rex 2454 df-v 2732 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-nul 3415 df-pw 3566 df-sn 3587 df-pr 3588 df-op 3590 df-uni 3795 df-br 3988 df-opab 4049 df-id 4276 df-xp 4615 df-rel 4616 df-cnv 4617 df-co 4618 df-dm 4619 df-rn 4620 df-res 4621 df-ima 4622 df-fun 5198 df-fn 5199 df-f 5200 df-f1 5201 df-fo 5202 df-f1o 5203 df-en 6716 df-dom 6717 |
This theorem is referenced by: (None) |
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