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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 37114 | . 2 ⊢ (𝐴 ∈ V → {𝐴} ∈ Moore) | |
| 2 | snprc 4717 | . . . . 5 ⊢ (¬ 𝐴 ∈ V ↔ {𝐴} = ∅) | |
| 3 | 2 | biimpi 216 | . . . 4 ⊢ (¬ 𝐴 ∈ V → {𝐴} = ∅) | 
| 4 | bj-0nmoore 37113 | . . . . 5 ⊢ ¬ ∅ ∈ Moore | |
| 5 | 4 | a1i 11 | . . . 4 ⊢ (¬ 𝐴 ∈ V → ¬ ∅ ∈ Moore) | 
| 6 | 3, 5 | eqneltrd 2861 | . . 3 ⊢ (¬ 𝐴 ∈ V → ¬ {𝐴} ∈ Moore) | 
| 7 | 6 | con4i 114 | . 2 ⊢ ({𝐴} ∈ Moore → 𝐴 ∈ V) | 
| 8 | 1, 7 | impbii 209 | 1 ⊢ (𝐴 ∈ V ↔ {𝐴} ∈ Moore) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ¬ wn 3 ↔ wb 206 = wceq 1540 ∈ wcel 2108 Vcvv 3480 ∅c0 4333 {csn 4626 Moorecmoore 37104 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-pw 4602 df-sn 4627 df-pr 4629 df-uni 4908 df-int 4947 df-bj-moore 37105 | 
| This theorem is referenced by: (None) | 
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