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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 4158 | . 2 ⊢ (∅ ∪ 𝐴) = (𝐴 ∪ ∅) | |
| 2 | un0 4394 | . 2 ⊢ (𝐴 ∪ ∅) = 𝐴 | |
| 3 | 1, 2 | eqtri 2765 | 1 ⊢ (∅ ∪ 𝐴) = 𝐴 |
| Colors of variables: wff setvar class |
| Syntax hints: = wceq 1540 ∪ cun 3949 ∅c0 4333 |
| 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-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-v 3482 df-dif 3954 df-un 3956 df-nul 4334 |
| This theorem is referenced by: nlim2 8528 pwmndid 18949 pwmnd 18950 psdmullem 22169 sltlpss 27945 slelss 27946 mulsrid 28139 mulsproplem5 28146 mulsproplem6 28147 mulsproplem7 28148 mulsproplem8 28149 coprprop 32708 fzodif1 32794 cycpmrn 33163 bj-pr22val 37020 bj-snfromadj 37045 metakunt17 42222 tfsconcat0i 43358 fiiuncl 45070 founiiun0 45195 infxrpnf 45457 prsal 46333 meadjun 46477 caragenuncllem 46527 carageniuncllem1 46536 hoidmvle 46615 iscnrm3rlem1 48837 |
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