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Mirrors > Home > MPE Home > Th. List > 0un | Structured version Visualization version GIF version |
Description: The union of the empty set with a class is itself. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
Ref | Expression |
---|---|
0un | ⊢ (∅ ∪ 𝐴) = 𝐴 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | uncom 4150 | . 2 ⊢ (∅ ∪ 𝐴) = (𝐴 ∪ ∅) | |
2 | un0 4388 | . 2 ⊢ (𝐴 ∪ ∅) = 𝐴 | |
3 | 1, 2 | eqtri 2754 | 1 ⊢ (∅ ∪ 𝐴) = 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1534 ∪ cun 3944 ∅c0 4322 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-ext 2697 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-tru 1537 df-fal 1547 df-ex 1775 df-sb 2061 df-clab 2704 df-cleq 2718 df-clel 2803 df-v 3464 df-dif 3949 df-un 3951 df-nul 4323 |
This theorem is referenced by: nlim2 8512 pwmndid 18921 pwmnd 18922 psdmullem 22155 sltlpss 27927 slelss 27928 mulsrid 28111 mulsproplem5 28118 mulsproplem6 28119 mulsproplem7 28120 mulsproplem8 28121 coprprop 32611 fzodif1 32698 cycpmrn 33025 bj-pr22val 36739 bj-snfromadj 36764 metakunt17 41929 tfsconcat0i 43048 fiiuncl 44703 founiiun0 44833 infxrpnf 45097 prsal 45975 meadjun 46119 caragenuncllem 46169 carageniuncllem1 46178 hoidmvle 46257 iscnrm3rlem1 48310 |
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