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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 4129 | . 2 ⊢ (∅ ∪ 𝐴) = (𝐴 ∪ ∅) | |
2 | un0 4344 | . 2 ⊢ (𝐴 ∪ ∅) = 𝐴 | |
3 | 1, 2 | eqtri 2844 | 1 ⊢ (∅ ∪ 𝐴) = 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 ∪ cun 3934 ∅c0 4291 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-v 3496 df-dif 3939 df-un 3941 df-nul 4292 |
This theorem is referenced by: pwmndid 18101 pwmnd 18102 coprprop 30435 fzodif1 30516 cycpmrn 30785 bj-pr22val 34334 fiiuncl 41347 founiiun0 41471 infxrpnf 41741 prsal 42623 meadjun 42764 caragenuncllem 42814 carageniuncllem1 42823 hoidmvle 42902 |
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