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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-snmooreb | Structured version Visualization version GIF version |
Description: A singleton is a Moore collection, biconditional version. (Contributed by BJ, 9-Dec-2021.) (Proof shortened by BJ, 10-Apr-2024.) |
Ref | Expression |
---|---|
bj-snmooreb | ⊢ (𝐴 ∈ V ↔ {𝐴} ∈ Moore) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bj-snmoore 34976 | . 2 ⊢ (𝐴 ∈ V → {𝐴} ∈ Moore) | |
2 | snprc 4623 | . . . . 5 ⊢ (¬ 𝐴 ∈ V ↔ {𝐴} = ∅) | |
3 | 2 | biimpi 219 | . . . 4 ⊢ (¬ 𝐴 ∈ V → {𝐴} = ∅) |
4 | bj-0nmoore 34975 | . . . . 5 ⊢ ¬ ∅ ∈ Moore | |
5 | 4 | a1i 11 | . . . 4 ⊢ (¬ 𝐴 ∈ V → ¬ ∅ ∈ Moore) |
6 | 3, 5 | eqneltrd 2853 | . . 3 ⊢ (¬ 𝐴 ∈ V → ¬ {𝐴} ∈ Moore) |
7 | 6 | con4i 114 | . 2 ⊢ ({𝐴} ∈ Moore → 𝐴 ∈ V) |
8 | 1, 7 | impbii 212 | 1 ⊢ (𝐴 ∈ V ↔ {𝐴} ∈ Moore) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 209 = wceq 1543 ∈ wcel 2110 Vcvv 3401 ∅c0 4227 {csn 4531 Moorecmoore 34966 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-ext 2706 ax-sep 5181 ax-nul 5188 ax-pow 5247 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-tru 1546 df-fal 1556 df-ex 1788 df-sb 2071 df-clab 2713 df-cleq 2726 df-clel 2812 df-ne 2936 df-ral 3059 df-rex 3060 df-rab 3063 df-v 3403 df-dif 3860 df-un 3862 df-in 3864 df-ss 3874 df-nul 4228 df-pw 4505 df-sn 4532 df-pr 4534 df-uni 4810 df-int 4850 df-bj-moore 34967 |
This theorem is referenced by: (None) |
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