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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 7942. (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 7489 | . . 3 ⊢ (𝒫 ∪ 𝐴 ∈ 𝐴 → ∪ 𝐴 ∈ V) | |
2 | pwuninel2 7942 | . . 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 184 | 1 ⊢ ¬ 𝒫 ∪ 𝐴 ∈ 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∈ wcel 2114 Vcvv 3496 𝒫 cpw 4541 ∪ cuni 4840 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pr 5332 ax-un 7463 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-rab 3149 df-v 3498 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-pw 4543 df-sn 4570 df-pr 4572 df-uni 4841 |
This theorem is referenced by: undefnel2 7945 disjen 8676 pnfnre 10684 kelac2lem 39671 kelac2 39672 ndfatafv2nrn 43427 afv2ndefb 43430 |
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