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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 5444. The proof of this theorem was automatically generated from unipwrVD 43428 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 3478 | . . . 4 ⊢ 𝑥 ∈ V | |
2 | 1 | snid 4659 | . . 3 ⊢ 𝑥 ∈ {𝑥} |
3 | snelpwi 5437 | . . 3 ⊢ (𝑥 ∈ 𝐴 → {𝑥} ∈ 𝒫 𝐴) | |
4 | elunii 4907 | . . 3 ⊢ ((𝑥 ∈ {𝑥} ∧ {𝑥} ∈ 𝒫 𝐴) → 𝑥 ∈ ∪ 𝒫 𝐴) | |
5 | 2, 3, 4 | sylancr 587 | . 2 ⊢ (𝑥 ∈ 𝐴 → 𝑥 ∈ ∪ 𝒫 𝐴) |
6 | 5 | ssriv 3983 | 1 ⊢ 𝐴 ⊆ ∪ 𝒫 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2106 ⊆ wss 3945 𝒫 cpw 4597 {csn 4623 ∪ cuni 4902 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2703 ax-sep 5293 ax-pr 5421 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-tru 1544 df-ex 1782 df-sb 2068 df-clab 2710 df-cleq 2724 df-clel 2810 df-v 3476 df-un 3950 df-in 3952 df-ss 3962 df-pw 4599 df-sn 4624 df-pr 4626 df-uni 4903 |
This theorem is referenced by: (None) |
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