![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 4290 | . . . 4 ⊢ ¬ 〈𝑥, 𝑦〉 ∈ ∅ | |
2 | 1 | nex 1802 | . . 3 ⊢ ¬ ∃𝑦〈𝑥, 𝑦〉 ∈ ∅ |
3 | vex 3449 | . . . 4 ⊢ 𝑥 ∈ V | |
4 | 3 | eldm2 5857 | . . 3 ⊢ (𝑥 ∈ dom ∅ ↔ ∃𝑦〈𝑥, 𝑦〉 ∈ ∅) |
5 | 2, 4 | mtbir 322 | . 2 ⊢ ¬ 𝑥 ∈ dom ∅ |
6 | 5 | nel0 4310 | 1 ⊢ dom ∅ = ∅ |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1541 ∃wex 1781 ∈ wcel 2106 ∅c0 4282 〈cop 4592 dom cdm 5633 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2707 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-sb 2068 df-clab 2714 df-cleq 2728 df-clel 2814 df-rab 3408 df-v 3447 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-nul 4283 df-if 4487 df-sn 4587 df-pr 4589 df-op 4593 df-br 5106 df-dm 5643 |
This theorem is referenced by: rn0 5881 dmxpid 5885 dmxpss 6123 fn0 6632 f0dom0 6726 f10d 6818 f1o00 6819 0fv 6886 1stval 7922 bropopvvv 8021 bropfvvvv 8023 supp0 8096 tz7.44lem1 8350 tz7.44-2 8352 tz7.44-3 8353 oicl 9464 oif 9465 swrd0 14545 dmtrclfv 14902 relexpdmd 14928 symgsssg 19247 symgfisg 19248 psgnunilem5 19274 dvbsss 25264 perfdvf 25265 uhgr0e 28020 uhgr0 28022 usgr0 28189 egrsubgr 28223 0grsubgr 28224 vtxdg0e 28420 eupth0 29156 dmadjrnb 30846 eldmne0 31540 f1ocnt 31700 tocyccntz 31988 mbfmcst 32850 0rrv 33042 matunitlindf 36067 ismgmOLD 36300 conrel2d 41918 neicvgbex 42366 iblempty 44178 |
Copyright terms: Public domain | W3C validator |