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| Mirrors > Home > MPE Home > Th. List > dm0 | Structured version Visualization version GIF 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 | ⊢ dom ∅ = ∅ |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | noel 4338 | . . . 4 ⊢ ¬ 〈𝑥, 𝑦〉 ∈ ∅ | |
| 2 | 1 | nex 1800 | . . 3 ⊢ ¬ ∃𝑦〈𝑥, 𝑦〉 ∈ ∅ |
| 3 | vex 3484 | . . . 4 ⊢ 𝑥 ∈ V | |
| 4 | 3 | eldm2 5912 | . . 3 ⊢ (𝑥 ∈ dom ∅ ↔ ∃𝑦〈𝑥, 𝑦〉 ∈ ∅) |
| 5 | 2, 4 | mtbir 323 | . 2 ⊢ ¬ 𝑥 ∈ dom ∅ |
| 6 | 5 | nel0 4354 | 1 ⊢ dom ∅ = ∅ |
| Colors of variables: wff setvar class |
| Syntax hints: = wceq 1540 ∃wex 1779 ∈ wcel 2108 ∅c0 4333 〈cop 4632 dom cdm 5685 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-br 5144 df-dm 5695 |
| This theorem is referenced by: rn0 5936 dmxpid 5941 dmxpss 6191 fn0 6699 f0dom0 6792 f10d 6882 f1o00 6883 0fv 6950 1stval 8016 bropopvvv 8115 bropfvvvv 8117 supp0 8190 tz7.44lem1 8445 tz7.44-2 8447 tz7.44-3 8448 oicl 9569 oif 9570 swrd0 14696 dmtrclfv 15057 relexpdmd 15083 symgsssg 19485 symgfisg 19486 psgnunilem5 19512 dvbsss 25937 perfdvf 25938 uhgr0e 29088 uhgr0 29090 usgr0 29260 egrsubgr 29294 0grsubgr 29295 vtxdg0e 29492 eupth0 30233 dmadjrnb 31925 eldmne0 32638 of0r 32688 f1ocnt 32804 tocyccntz 33164 mbfmcst 34261 0rrv 34453 matunitlindf 37625 ismgmOLD 37857 conrel2d 43677 neicvgbex 44125 iblempty 45980 |
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