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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 4613 | . . 3 ⊢ (¬ 𝐴 ∈ V ↔ {𝐴} = ∅) | |
2 | df-ne 2988 | . . . 4 ⊢ ({𝐴} ≠ ∅ ↔ ¬ {𝐴} = ∅) | |
3 | 2 | con2bii 361 | . . 3 ⊢ ({𝐴} = ∅ ↔ ¬ {𝐴} ≠ ∅) |
4 | 1, 3 | bitri 278 | . 2 ⊢ (¬ 𝐴 ∈ V ↔ ¬ {𝐴} ≠ ∅) |
5 | 4 | con4bii 324 | 1 ⊢ (𝐴 ∈ V ↔ {𝐴} ≠ ∅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 209 = wceq 1538 ∈ wcel 2111 ≠ wne 2987 Vcvv 3441 ∅c0 4243 {csn 4525 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-ext 2770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-ex 1782 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-ne 2988 df-v 3443 df-dif 3884 df-nul 4244 df-sn 4526 |
This theorem is referenced by: lpvtx 26861 loop1cycl 32497 elima4 33132 |
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