![]() |
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 35515 | . 2 ⊢ (𝐴 ∈ V → {𝐴} ∈ Moore) | |
2 | snprc 4676 | . . . . 5 ⊢ (¬ 𝐴 ∈ V ↔ {𝐴} = ∅) | |
3 | 2 | biimpi 215 | . . . 4 ⊢ (¬ 𝐴 ∈ V → {𝐴} = ∅) |
4 | bj-0nmoore 35514 | . . . . 5 ⊢ ¬ ∅ ∈ Moore | |
5 | 4 | a1i 11 | . . . 4 ⊢ (¬ 𝐴 ∈ V → ¬ ∅ ∈ Moore) |
6 | 3, 5 | eqneltrd 2857 | . . 3 ⊢ (¬ 𝐴 ∈ V → ¬ {𝐴} ∈ Moore) |
7 | 6 | con4i 114 | . 2 ⊢ ({𝐴} ∈ Moore → 𝐴 ∈ V) |
8 | 1, 7 | impbii 208 | 1 ⊢ (𝐴 ∈ V ↔ {𝐴} ∈ Moore) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 205 = wceq 1541 ∈ wcel 2106 Vcvv 3443 ∅c0 4280 {csn 4584 Moorecmoore 35505 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2708 ax-sep 5254 ax-nul 5261 ax-pow 5318 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-tru 1544 df-fal 1554 df-ex 1782 df-sb 2068 df-clab 2715 df-cleq 2729 df-clel 2815 df-ne 2942 df-ral 3063 df-rex 3072 df-rab 3406 df-v 3445 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-nul 4281 df-pw 4560 df-sn 4585 df-pr 4587 df-uni 4864 df-int 4906 df-bj-moore 35506 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |