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Mirrors > Home > MPE Home > Th. List > Mathboxes > unipwr | Structured version Visualization version GIF version |
Description: A class is a subclass of the union of its power class. This theorem is the right-to-left subclass lemma of unipw 5306. The proof of this theorem was automatically generated from unipwrVD 41974 using a tools command file , translateMWO.cmd , by translating the proof into its non-virtual deduction form and minimizing it. (Contributed by Alan Sare, 25-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
unipwr | ⊢ 𝐴 ⊆ ∪ 𝒫 𝐴 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vex 3401 | . . . 4 ⊢ 𝑥 ∈ V | |
2 | 1 | snid 4549 | . . 3 ⊢ 𝑥 ∈ {𝑥} |
3 | snelpwi 5300 | . . 3 ⊢ (𝑥 ∈ 𝐴 → {𝑥} ∈ 𝒫 𝐴) | |
4 | elunii 4798 | . . 3 ⊢ ((𝑥 ∈ {𝑥} ∧ {𝑥} ∈ 𝒫 𝐴) → 𝑥 ∈ ∪ 𝒫 𝐴) | |
5 | 2, 3, 4 | sylancr 590 | . 2 ⊢ (𝑥 ∈ 𝐴 → 𝑥 ∈ ∪ 𝒫 𝐴) |
6 | 5 | ssriv 3879 | 1 ⊢ 𝐴 ⊆ ∪ 𝒫 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2113 ⊆ wss 3841 𝒫 cpw 4485 {csn 4513 ∪ cuni 4793 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1916 ax-6 1974 ax-7 2019 ax-8 2115 ax-9 2123 ax-ext 2710 ax-sep 5164 ax-nul 5171 ax-pr 5293 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-tru 1545 df-fal 1555 df-ex 1787 df-sb 2074 df-clab 2717 df-cleq 2730 df-clel 2811 df-v 3399 df-dif 3844 df-un 3846 df-in 3848 df-ss 3858 df-nul 4210 df-pw 4487 df-sn 4514 df-pr 4516 df-uni 4794 |
This theorem is referenced by: (None) |
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