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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 4064 | . 2 ⊢ (∅ ∪ 𝐴) = (𝐴 ∪ ∅) | |
2 | un0 4302 | . 2 ⊢ (𝐴 ∪ ∅) = 𝐴 | |
3 | 1, 2 | eqtri 2765 | 1 ⊢ (∅ ∪ 𝐴) = 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1543 ∪ cun 3861 ∅c0 4234 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-ext 2708 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-tru 1546 df-fal 1556 df-ex 1788 df-sb 2071 df-clab 2715 df-cleq 2729 df-clel 2816 df-v 3407 df-dif 3866 df-un 3868 df-nul 4235 |
This theorem is referenced by: pwmndid 18360 pwmnd 18361 coprprop 30749 fzodif1 30831 cycpmrn 31126 sltlpss 33821 bj-pr22val 34943 metakunt17 39861 fiiuncl 42284 founiiun0 42399 infxrpnf 42658 prsal 43532 meadjun 43673 caragenuncllem 43723 carageniuncllem1 43732 hoidmvle 43811 iscnrm3rlem1 45905 |
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