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Mirrors > Home > MPE Home > Th. List > snn0d | Structured version Visualization version GIF version |
Description: The singleton of a set is not empty. (Contributed by Glauco Siliprandi, 3-Mar-2021.) |
Ref | Expression |
---|---|
snn0d.1 | ⊢ (𝜑 → 𝐴 ∈ 𝑉) |
Ref | Expression |
---|---|
snn0d | ⊢ (𝜑 → {𝐴} ≠ ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | snn0d.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
2 | snnzg 4779 | . 2 ⊢ (𝐴 ∈ 𝑉 → {𝐴} ≠ ∅) | |
3 | 1, 2 | syl 17 | 1 ⊢ (𝜑 → {𝐴} ≠ ∅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2106 ≠ wne 2938 ∅c0 4339 {csn 4631 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-ne 2939 df-dif 3966 df-nul 4340 df-sn 4632 |
This theorem is referenced by: 0nelop 5506 rnglidl0 21257 hausflim 24005 flimcf 24006 flimclslem 24008 cnpflf2 24024 cnpflf 24025 neipcfilu 24321 zarclssn 33834 zar0ring 33839 elpaddat 39787 mnuprdlem1 44268 difmapsn 45155 ovnovollem1 46612 ovnovollem3 46614 |
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