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Mathbox for Glauco Siliprandi |
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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 4002 | . 2 ⊢ {∅} ⊆ {∅} | |
2 | uni0b 4933 | . 2 ⊢ (∪ {∅} = ∅ ↔ {∅} ⊆ {∅}) | |
3 | 1, 2 | mpbir 230 | 1 ⊢ ∪ {∅} = ∅ |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1542 ⊆ wss 3946 ∅c0 4320 {csn 4624 ∪ cuni 4904 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-11 2155 ax-ext 2704 |
This theorem depends on definitions: df-bi 206 df-an 398 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-ral 3063 df-rex 3072 df-v 3477 df-dif 3949 df-in 3953 df-ss 3963 df-nul 4321 df-sn 4625 df-uni 4905 |
This theorem is referenced by: founiiun0 43759 prsal 44907 |
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