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Mirrors > Home > MPE Home > Th. List > snexg | Structured version Visualization version GIF version |
Description: A singleton built on a set is a set. Special case of snex 5427 which does not require ax-nul 5302 and is intuitionistically valid. (Contributed by NM, 7-Aug-1994.) (Revised by Mario Carneiro, 19-May-2013.) Extract from snex 5427 and shorten proof. (Revised by BJ, 15-Jan-2025.) |
Ref | Expression |
---|---|
snexg | ⊢ (𝐴 ∈ 𝑉 → {𝐴} ∈ V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sneq 4634 | . . 3 ⊢ (𝑥 = 𝐴 → {𝑥} = {𝐴}) | |
2 | vsnex 5425 | . . 3 ⊢ {𝑥} ∈ V | |
3 | 1, 2 | eqeltrrdi 2843 | . 2 ⊢ (𝑥 = 𝐴 → {𝐴} ∈ V) |
4 | 3 | vtocleg 3544 | 1 ⊢ (𝐴 ∈ 𝑉 → {𝐴} ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2107 Vcvv 3475 {csn 4624 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 ax-sep 5295 ax-pr 5423 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-tru 1545 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-v 3477 df-un 3951 df-sn 4625 df-pr 4627 |
This theorem is referenced by: snex 5427 selsALT 5435 snelpwg 5438 intidg 5453 |
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