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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 3992 | . 2 ⊢ {∅} ⊆ {∅} | |
2 | uni0b 4867 | . 2 ⊢ (∪ {∅} = ∅ ↔ {∅} ⊆ {∅}) | |
3 | 1, 2 | mpbir 233 | 1 ⊢ ∪ {∅} = ∅ |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1536 ⊆ wss 3939 ∅c0 4294 {csn 4570 ∪ cuni 4841 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ral 3146 df-rex 3147 df-v 3499 df-dif 3942 df-in 3946 df-ss 3955 df-nul 4295 df-sn 4571 df-uni 4842 |
This theorem is referenced by: founiiun0 41457 prsal 42610 |
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