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Mirrors > Home > MPE Home > Th. List > Mathboxes > 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 4710 | . 2 ⊢ (𝐴 ∈ 𝑉 → {𝐴} ≠ ∅) | |
3 | 1, 2 | syl 17 | 1 ⊢ (𝜑 → {𝐴} ≠ ∅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2114 ≠ wne 3016 ∅c0 4291 {csn 4567 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-dif 3939 df-nul 4292 df-sn 4568 |
This theorem is referenced by: difmapsn 41495 ovnovollem1 42958 |
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