![]() |
Mathbox for BJ |
< Previous
Next >
Nearby theorems |
|
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 37096 | . 2 ⊢ (𝐴 ∈ V → {𝐴} ∈ Moore) | |
2 | snprc 4722 | . . . . 5 ⊢ (¬ 𝐴 ∈ V ↔ {𝐴} = ∅) | |
3 | 2 | biimpi 216 | . . . 4 ⊢ (¬ 𝐴 ∈ V → {𝐴} = ∅) |
4 | bj-0nmoore 37095 | . . . . 5 ⊢ ¬ ∅ ∈ Moore | |
5 | 4 | a1i 11 | . . . 4 ⊢ (¬ 𝐴 ∈ V → ¬ ∅ ∈ Moore) |
6 | 3, 5 | eqneltrd 2859 | . . 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 1537 ∈ wcel 2106 Vcvv 3478 ∅c0 4339 {csn 4631 Moorecmoore 37086 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-pw 4607 df-sn 4632 df-pr 4634 df-uni 4913 df-int 4952 df-bj-moore 37087 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |