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| Mirrors > Home > MPE Home > Th. List > pwuninel | Structured version Visualization version GIF version | ||
| Description: The powerclass of the union of a class does not belong to that class. This theorem provides a way of constructing a new set that does not belong to a given set. See also pwuninel2 8299. (Contributed by NM, 27-Jun-2008.) (Proof shortened by Mario Carneiro, 23-Dec-2016.) | 
| Ref | Expression | 
|---|---|
| pwuninel | ⊢ ¬ 𝒫 ∪ 𝐴 ∈ 𝐴 | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | pwexr 7785 | . . 3 ⊢ (𝒫 ∪ 𝐴 ∈ 𝐴 → ∪ 𝐴 ∈ V) | |
| 2 | pwuninel2 8299 | . . 3 ⊢ (∪ 𝐴 ∈ V → ¬ 𝒫 ∪ 𝐴 ∈ 𝐴) | |
| 3 | 1, 2 | syl 17 | . 2 ⊢ (𝒫 ∪ 𝐴 ∈ 𝐴 → ¬ 𝒫 ∪ 𝐴 ∈ 𝐴) | 
| 4 | id 22 | . 2 ⊢ (¬ 𝒫 ∪ 𝐴 ∈ 𝐴 → ¬ 𝒫 ∪ 𝐴 ∈ 𝐴) | |
| 5 | 3, 4 | pm2.61i 182 | 1 ⊢ ¬ 𝒫 ∪ 𝐴 ∈ 𝐴 | 
| Colors of variables: wff setvar class | 
| Syntax hints: ¬ wn 3 ∈ wcel 2108 Vcvv 3480 𝒫 cpw 4600 ∪ cuni 4907 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-sep 5296 ax-pr 5432 ax-un 7755 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-rab 3437 df-v 3482 df-un 3956 df-in 3958 df-ss 3968 df-pw 4602 df-sn 4627 df-pr 4629 df-uni 4908 | 
| This theorem is referenced by: undefnel2 8302 disjen 9174 pnfnre 11302 kelac2lem 43076 kelac2 43077 ndfatafv2nrn 47233 afv2ndefb 47236 | 
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