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Mirrors > Home > ILE Home > Th. List > rn0 | GIF version |
Description: The range of the empty set is empty. Part of Theorem 3.8(v) of [Monk1] p. 36. (Contributed by NM, 4-Jul-1994.) |
Ref | Expression |
---|---|
rn0 | ⊢ ran ∅ = ∅ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dm0 4859 | . 2 ⊢ dom ∅ = ∅ | |
2 | dm0rn0 4862 | . 2 ⊢ (dom ∅ = ∅ ↔ ran ∅ = ∅) | |
3 | 1, 2 | mpbi 145 | 1 ⊢ ran ∅ = ∅ |
Colors of variables: wff set class |
Syntax hints: = wceq 1364 ∅c0 3437 dom cdm 4644 ran crn 4645 |
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-14 2163 ax-ext 2171 ax-sep 4136 ax-pow 4192 ax-pr 4227 |
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 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-v 2754 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-nul 3438 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-br 4019 df-opab 4080 df-cnv 4652 df-dm 4654 df-rn 4655 |
This theorem is referenced by: ima0 5005 0ima 5006 xpima1 5093 f0 5425 exmidfodomrlemim 7230 |
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