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| Mirrors > Home > MPE Home > Th. List > Mathboxes > unisn0 | Structured version Visualization version GIF version | ||
| Description: The union of the singleton of the empty set is the empty set. (Contributed by Glauco Siliprandi, 17-Aug-2020.) | 
| Ref | Expression | 
|---|---|
| unisn0 | ⊢ ∪ {∅} = ∅ | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | ssid 4005 | . 2 ⊢ {∅} ⊆ {∅} | |
| 2 | uni0b 4932 | . 2 ⊢ (∪ {∅} = ∅ ↔ {∅} ⊆ {∅}) | |
| 3 | 1, 2 | mpbir 231 | 1 ⊢ ∪ {∅} = ∅ | 
| Colors of variables: wff setvar class | 
| Syntax hints: = wceq 1539 ⊆ wss 3950 ∅c0 4332 {csn 4625 ∪ cuni 4906 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-11 2156 ax-ext 2707 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1542 df-fal 1552 df-ex 1779 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-ral 3061 df-rex 3070 df-v 3481 df-dif 3953 df-ss 3967 df-nul 4333 df-sn 4626 df-uni 4907 | 
| This theorem is referenced by: founiiun0 45200 prsal 46338 | 
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