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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 3456 |
. 2
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2 | noel 3441 |
. . . 4
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3 | 2 | nex 1511 |
. . 3
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4 | vex 2755 |
. . . 4
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5 | 4 | eldm2 4843 |
. . 3
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6 | 3, 5 | mtbir 672 |
. 2
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7 | 1, 6 | mpgbir 1464 |
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-ext 2171 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-v 2754 df-dif 3146 df-un 3148 df-nul 3438 df-sn 3613 df-pr 3614 df-op 3616 df-br 4019 df-dm 4654 |
This theorem is referenced by: rn0 4901 sqxpeq0 5070 fn0 5354 f0dom0 5428 f1o00 5515 rdg0 6413 frec0g 6423 ennnfonelemj0 12455 ennnfonelem1 12461 ennnfonelemkh 12466 ennnfonelemhf1o 12467 |
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