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| Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-snexg | Structured version Visualization version GIF version | ||
| Description: A singleton built on a set is a set. Contrary to bj-snex 37036, this proof is intuitionistically valid and does not require ax-nul 5306. (Contributed by NM, 7-Aug-1994.) Extract it from snex 5436 and prove it from ax-bj-sn 37034. (Revised by BJ, 12-Jan-2025.) (Proof modification is discouraged.) | 
| Ref | Expression | 
|---|---|
| bj-snexg | ⊢ (𝐴 ∈ 𝑉 → {𝐴} ∈ V) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | sneq 4636 | . . 3 ⊢ (𝑥 = 𝐴 → {𝑥} = {𝐴}) | |
| 2 | ax-bj-sn 37034 | . . . . 5 ⊢ ∀𝑥∃𝑦∀𝑧(𝑧 ∈ 𝑦 ↔ 𝑧 = 𝑥) | |
| 3 | 2 | spi 2184 | . . . 4 ⊢ ∃𝑦∀𝑧(𝑧 ∈ 𝑦 ↔ 𝑧 = 𝑥) | 
| 4 | bj-axsn 37033 | . . . 4 ⊢ ({𝑥} ∈ V ↔ ∃𝑦∀𝑧(𝑧 ∈ 𝑦 ↔ 𝑧 = 𝑥)) | |
| 5 | 3, 4 | mpbir 231 | . . 3 ⊢ {𝑥} ∈ V | 
| 6 | 1, 5 | eqeltrrdi 2850 | . 2 ⊢ (𝑥 = 𝐴 → {𝐴} ∈ V) | 
| 7 | 6 | vtocleg 3553 | 1 ⊢ (𝐴 ∈ 𝑉 → {𝐴} ∈ V) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 ∀wal 1538 = wceq 1540 ∃wex 1779 ∈ wcel 2108 Vcvv 3480 {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-bj-sn 37034 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1543 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-v 3482 df-sn 4627 | 
| This theorem is referenced by: bj-snex 37036 bj-prexg 37040 bj-adjfrombun 37047 | 
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