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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 34429 | . 2 ⊢ (𝐴 ∈ V → {𝐴} ∈ Moore) | |
2 | snprc 4646 | . . . . 5 ⊢ (¬ 𝐴 ∈ V ↔ {𝐴} = ∅) | |
3 | 2 | biimpi 218 | . . . 4 ⊢ (¬ 𝐴 ∈ V → {𝐴} = ∅) |
4 | bj-0nmoore 34428 | . . . . 5 ⊢ ¬ ∅ ∈ Moore | |
5 | 4 | a1i 11 | . . . 4 ⊢ (¬ 𝐴 ∈ V → ¬ ∅ ∈ Moore) |
6 | 3, 5 | eqneltrd 2931 | . . 3 ⊢ (¬ 𝐴 ∈ V → ¬ {𝐴} ∈ Moore) |
7 | 6 | con4i 114 | . 2 ⊢ ({𝐴} ∈ Moore → 𝐴 ∈ V) |
8 | 1, 7 | impbii 211 | 1 ⊢ (𝐴 ∈ V ↔ {𝐴} ∈ Moore) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 208 = wceq 1536 ∈ wcel 2113 Vcvv 3491 ∅c0 4284 {csn 4560 Moorecmoore 34419 |
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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 ax-sep 5196 ax-nul 5203 ax-pow 5259 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1084 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ne 3016 df-ral 3142 df-rex 3143 df-rab 3146 df-v 3493 df-dif 3932 df-un 3934 df-in 3936 df-ss 3945 df-nul 4285 df-pw 4534 df-sn 4561 df-pr 4563 df-uni 4832 df-int 4870 df-bj-moore 34420 |
This theorem is referenced by: (None) |
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