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| Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-snex | Structured version Visualization version GIF version | ||
| Description: A singleton is a set. See also snex 5436, snexALT 5383. (Contributed by NM, 7-Aug-1994.) Prove it from ax-bj-sn 37034. (Revised by BJ, 12-Jan-2025.) (Proof modification is discouraged.) |
| Ref | Expression |
|---|---|
| bj-snex | ⊢ {𝐴} ∈ V |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | bj-snexg 37035 | . 2 ⊢ (𝐴 ∈ V → {𝐴} ∈ V) | |
| 2 | snprc 4717 | . . . 4 ⊢ (¬ 𝐴 ∈ V ↔ {𝐴} = ∅) | |
| 3 | 2 | biimpi 216 | . . 3 ⊢ (¬ 𝐴 ∈ V → {𝐴} = ∅) |
| 4 | 0ex 5307 | . . 3 ⊢ ∅ ∈ V | |
| 5 | 3, 4 | eqeltrdi 2849 | . 2 ⊢ (¬ 𝐴 ∈ V → {𝐴} ∈ V) |
| 6 | 1, 5 | pm2.61i 182 | 1 ⊢ {𝐴} ∈ V |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 = wceq 1540 ∈ wcel 2108 Vcvv 3480 ∅c0 4333 {csn 4626 |
| 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-12 2177 ax-ext 2708 ax-nul 5306 ax-bj-sn 37034 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-v 3482 df-dif 3954 df-nul 4334 df-sn 4627 |
| This theorem is referenced by: bj-prex 37041 |
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