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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 4676 | . 2 ⊢ (𝐴 ∈ 𝑉 → {𝐴} ≠ ∅) | |
3 | 1, 2 | syl 17 | 1 ⊢ (𝜑 → {𝐴} ≠ ∅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2112 ≠ wne 2932 ∅c0 4223 {csn 4527 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-ext 2708 |
This theorem depends on definitions: df-bi 210 df-an 400 df-tru 1546 df-fal 1556 df-ex 1788 df-sb 2073 df-clab 2715 df-cleq 2728 df-clel 2809 df-ne 2933 df-dif 3856 df-nul 4224 df-sn 4528 |
This theorem is referenced by: 0nelop 5364 hausflim 22832 flimcf 22833 flimclslem 22835 cnpflf2 22851 cnpflf 22852 neipcfilu 23147 zarclssn 31491 zar0ring 31496 elpaddat 37504 mnuprdlem1 41504 difmapsn 42366 ovnovollem1 43812 ovnovollem3 43814 |
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