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Mirrors > Home > MPE Home > Th. List > pwuninel | Structured version Visualization version GIF version |
Description: The power set of the union of a set does not belong to the set. This theorem provides a way of constructing a new set that doesn't belong to a given set. See also pwuninel2 8078. (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 7606 | . . 3 ⊢ (𝒫 ∪ 𝐴 ∈ 𝐴 → ∪ 𝐴 ∈ V) | |
2 | pwuninel2 8078 | . . 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 2106 Vcvv 3430 𝒫 cpw 4534 ∪ cuni 4840 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2709 ax-sep 5222 ax-nul 5229 ax-pr 5351 ax-un 7579 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-tru 1542 df-fal 1552 df-ex 1783 df-sb 2068 df-clab 2716 df-cleq 2730 df-clel 2816 df-rab 3073 df-v 3432 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-nul 4258 df-pw 4536 df-sn 4563 df-pr 4565 df-uni 4841 |
This theorem is referenced by: undefnel2 8081 disjen 8909 pnfnre 11004 kelac2lem 40875 kelac2 40876 ndfatafv2nrn 44669 afv2ndefb 44672 |
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