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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 2937 | . . . 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 205 = wceq 1534 ∈ wcel 2099 ≠ wne 2936 Vcvv 3470 ∅c0 4318 {csn 4624 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-ext 2699 |
This theorem depends on definitions: df-bi 206 df-an 396 df-tru 1537 df-fal 1547 df-ex 1775 df-sb 2061 df-clab 2706 df-cleq 2720 df-clel 2806 df-ne 2937 df-v 3472 df-dif 3948 df-nul 4319 df-sn 4625 |
This theorem is referenced by: lpvtx 28874 loop1cycl 34741 elima4 35365 |
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