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Mirrors > Home > MPE Home > Th. List > snelpw | Structured version Visualization version GIF version |
Description: A singleton of a set belongs to the power class of a class containing the set. (Contributed by NM, 1-Apr-1998.) |
Ref | Expression |
---|---|
snelpw.1 | ⊢ 𝐴 ∈ V |
Ref | Expression |
---|---|
snelpw | ⊢ (𝐴 ∈ 𝐵 ↔ {𝐴} ∈ 𝒫 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | snelpw.1 | . . 3 ⊢ 𝐴 ∈ V | |
2 | 1 | snss 4711 | . 2 ⊢ (𝐴 ∈ 𝐵 ↔ {𝐴} ⊆ 𝐵) |
3 | snex 5323 | . . 3 ⊢ {𝐴} ∈ V | |
4 | 3 | elpw 4545 | . 2 ⊢ ({𝐴} ∈ 𝒫 𝐵 ↔ {𝐴} ⊆ 𝐵) |
5 | 2, 4 | bitr4i 280 | 1 ⊢ (𝐴 ∈ 𝐵 ↔ {𝐴} ∈ 𝒫 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 ∈ wcel 2110 Vcvv 3494 ⊆ wss 3935 𝒫 cpw 4538 {csn 4560 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pr 5321 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-v 3496 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-pw 4540 df-sn 4561 df-pr 4563 |
This theorem is referenced by: dis2ndc 22062 dislly 22099 |
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