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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 3916 | . 2 ⊢ {∅} ⊆ {∅} | |
2 | uni0b 4776 | . 2 ⊢ (∪ {∅} = ∅ ↔ {∅} ⊆ {∅}) | |
3 | 1, 2 | mpbir 232 | 1 ⊢ ∪ {∅} = ∅ |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1525 ⊆ wss 3865 ∅c0 4217 {csn 4478 ∪ cuni 4751 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1781 ax-4 1795 ax-5 1892 ax-6 1951 ax-7 1996 ax-8 2085 ax-9 2093 ax-10 2114 ax-11 2128 ax-12 2143 ax-ext 2771 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-tru 1528 df-ex 1766 df-nf 1770 df-sb 2045 df-clab 2778 df-cleq 2790 df-clel 2865 df-nfc 2937 df-ral 3112 df-rex 3113 df-v 3442 df-dif 3868 df-in 3872 df-ss 3880 df-nul 4218 df-sn 4479 df-uni 4752 |
This theorem is referenced by: founiiun0 41012 prsal 42167 |
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