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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 4054 | . 2 ⊢ (∅ ∪ 𝐴) = (𝐴 ∪ ∅) | |
2 | un0 4268 | . 2 ⊢ (𝐴 ∪ ∅) = 𝐴 | |
3 | 1, 2 | eqtri 2819 | 1 ⊢ (∅ ∪ 𝐴) = 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1522 ∪ cun 3861 ∅c0 4215 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1777 ax-4 1791 ax-5 1888 ax-6 1947 ax-7 1992 ax-8 2083 ax-9 2091 ax-10 2112 ax-11 2126 ax-12 2141 ax-ext 2769 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-tru 1525 df-ex 1762 df-nf 1766 df-sb 2043 df-clab 2776 df-cleq 2788 df-clel 2863 df-nfc 2935 df-v 3439 df-dif 3866 df-un 3868 df-nul 4216 |
This theorem is referenced by: coprprop 30128 fzodif1 30207 cycpmrn 30427 fiiuncl 40892 founiiun0 41017 infxrpnf 41289 prsal 42172 meadjun 42313 caragenuncllem 42363 carageniuncllem1 42372 hoidmvle 42451 |
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