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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 3939 | . 2 ⊢ {∅} ⊆ {∅} | |
2 | uni0b 4864 | . 2 ⊢ (∪ {∅} = ∅ ↔ {∅} ⊆ {∅}) | |
3 | 1, 2 | mpbir 230 | 1 ⊢ ∪ {∅} = ∅ |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1539 ⊆ wss 3883 ∅c0 4253 {csn 4558 ∪ cuni 4836 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-11 2156 ax-ext 2709 |
This theorem depends on definitions: df-bi 206 df-an 396 df-tru 1542 df-fal 1552 df-ex 1784 df-sb 2069 df-clab 2716 df-cleq 2730 df-clel 2817 df-ral 3068 df-rex 3069 df-v 3424 df-dif 3886 df-in 3890 df-ss 3900 df-nul 4254 df-sn 4559 df-uni 4837 |
This theorem is referenced by: founiiun0 42617 prsal 43749 |
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