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Mirrors > Home > ILE Home > Th. List > dm0 | Unicode version |
Description: The domain of the empty set is empty. Part of Theorem 3.8(v) of [Monk1] p. 36. (Contributed by NM, 4-Jul-1994.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) |
Ref | Expression |
---|---|
dm0 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eq0 3386 |
. 2
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2 | noel 3372 |
. . . 4
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3 | 2 | nex 1477 |
. . 3
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4 | vex 2692 |
. . . 4
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5 | 4 | eldm2 4745 |
. . 3
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6 | 3, 5 | mtbir 661 |
. 2
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7 | 1, 6 | mpgbir 1430 |
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 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-fal 1338 df-nf 1438 df-sb 1737 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-v 2691 df-dif 3078 df-un 3080 df-nul 3369 df-sn 3538 df-pr 3539 df-op 3541 df-br 3938 df-dm 4557 |
This theorem is referenced by: rn0 4803 sqxpeq0 4970 fn0 5250 f0dom0 5324 f1o00 5410 rdg0 6292 frec0g 6302 ennnfonelemj0 11950 ennnfonelem1 11956 ennnfonelemkh 11961 ennnfonelemhf1o 11962 |
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