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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 4714 | . . 3 ⊢ (¬ 𝐴 ∈ V ↔ {𝐴} = ∅) | |
2 | df-ne 2933 | . . . 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 1533 ∈ wcel 2098 ≠ wne 2932 Vcvv 3466 ∅c0 4315 {csn 4621 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2695 |
This theorem depends on definitions: df-bi 206 df-an 396 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2702 df-cleq 2716 df-clel 2802 df-ne 2933 df-v 3468 df-dif 3944 df-nul 4316 df-sn 4622 |
This theorem is referenced by: lpvtx 28800 loop1cycl 34619 elima4 35243 |
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