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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 3988 | . 2 ⊢ {∅} ⊆ {∅} | |
2 | uni0b 4857 | . 2 ⊢ (∪ {∅} = ∅ ↔ {∅} ⊆ {∅}) | |
3 | 1, 2 | mpbir 232 | 1 ⊢ ∪ {∅} = ∅ |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1528 ⊆ wss 3935 ∅c0 4290 {csn 4559 ∪ cuni 4832 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2793 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-v 3497 df-dif 3938 df-in 3942 df-ss 3951 df-nul 4291 df-sn 4560 df-uni 4833 |
This theorem is referenced by: founiiun0 41331 prsal 42484 |
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