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| Mirrors > Home > MPE Home > Th. List > snnzb | Structured version Visualization version GIF version | ||
| Description: A singleton is nonempty iff its argument is a set. (Contributed by Scott Fenton, 8-May-2018.) |
| Ref | Expression |
|---|---|
| snnzb | ⊢ (𝐴 ∈ V ↔ {𝐴} ≠ ∅) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | snprc 4717 | . . 3 ⊢ (¬ 𝐴 ∈ V ↔ {𝐴} = ∅) | |
| 2 | df-ne 2941 | . . . 4 ⊢ ({𝐴} ≠ ∅ ↔ ¬ {𝐴} = ∅) | |
| 3 | 2 | con2bii 357 | . . 3 ⊢ ({𝐴} = ∅ ↔ ¬ {𝐴} ≠ ∅) |
| 4 | 1, 3 | bitri 275 | . 2 ⊢ (¬ 𝐴 ∈ V ↔ ¬ {𝐴} ≠ ∅) |
| 5 | 4 | con4bii 321 | 1 ⊢ (𝐴 ∈ V ↔ {𝐴} ≠ ∅) |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 ↔ wb 206 = wceq 1540 ∈ wcel 2108 ≠ wne 2940 Vcvv 3480 ∅c0 4333 {csn 4626 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-v 3482 df-dif 3954 df-nul 4334 df-sn 4627 |
| This theorem is referenced by: lpvtx 29085 loop1cycl 35142 elima4 35776 |
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