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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 37018, this proof is intuitionistically valid and does not require ax-nul 5312. (Contributed by NM, 7-Aug-1994.) Extract it from snex 5442 and prove it from ax-bj-sn 37016. (Revised by BJ, 12-Jan-2025.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
bj-snexg | ⊢ (𝐴 ∈ 𝑉 → {𝐴} ∈ V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sneq 4641 | . . 3 ⊢ (𝑥 = 𝐴 → {𝑥} = {𝐴}) | |
2 | ax-bj-sn 37016 | . . . . 5 ⊢ ∀𝑥∃𝑦∀𝑧(𝑧 ∈ 𝑦 ↔ 𝑧 = 𝑥) | |
3 | 2 | spi 2182 | . . . 4 ⊢ ∃𝑦∀𝑧(𝑧 ∈ 𝑦 ↔ 𝑧 = 𝑥) |
4 | bj-axsn 37015 | . . . 4 ⊢ ({𝑥} ∈ V ↔ ∃𝑦∀𝑧(𝑧 ∈ 𝑦 ↔ 𝑧 = 𝑥)) | |
5 | 3, 4 | mpbir 231 | . . 3 ⊢ {𝑥} ∈ V |
6 | 1, 5 | eqeltrrdi 2848 | . 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 1535 = wceq 1537 ∃wex 1776 ∈ wcel 2106 Vcvv 3478 {csn 4631 |
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-12 2175 ax-ext 2706 ax-bj-sn 37016 |
This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1540 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-v 3480 df-sn 4632 |
This theorem is referenced by: bj-snex 37018 bj-prexg 37022 bj-adjfrombun 37029 |
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