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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 4197 | . . 3 ⊢ (¬ 𝐴 ∈ V ↔ {𝐴} = ∅) | |
2 | df-ne 2782 | . . . 4 ⊢ ({𝐴} ≠ ∅ ↔ ¬ {𝐴} = ∅) | |
3 | 2 | con2bii 346 | . . 3 ⊢ ({𝐴} = ∅ ↔ ¬ {𝐴} ≠ ∅) |
4 | 1, 3 | bitri 263 | . 2 ⊢ (¬ 𝐴 ∈ V ↔ ¬ {𝐴} ≠ ∅) |
5 | 4 | con4bii 310 | 1 ⊢ (𝐴 ∈ V ↔ {𝐴} ≠ ∅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 195 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 Vcvv 3173 ∅c0 3874 {csn 4125 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-v 3175 df-dif 3543 df-nul 3875 df-sn 4126 |
This theorem is referenced by: lpvtx 25734 elima4 30924 |
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