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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 4773 | . 2 ⊢ (𝐴 ∈ 𝑉 → {𝐴} ≠ ∅) | |
| 3 | 1, 2 | syl 17 | 1 ⊢ (𝜑 → {𝐴} ≠ ∅) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∈ wcel 2107 ≠ wne 2939 ∅c0 4332 {csn 4625 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-ext 2707 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1542 df-fal 1552 df-ex 1779 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-ne 2940 df-dif 3953 df-nul 4333 df-sn 4626 | 
| This theorem is referenced by: 0nelop 5500 rnglidl0 21240 hausflim 23990 flimcf 23991 flimclslem 23993 cnpflf2 24009 cnpflf 24010 neipcfilu 24306 zarclssn 33873 zar0ring 33878 elpaddat 39807 mnuprdlem1 44296 difmapsn 45222 ovnovollem1 46676 ovnovollem3 46678 | 
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