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Mirrors > Home > MPE Home > Th. List > pwnss | Structured version Visualization version GIF version |
Description: The power set of a set is never a subset. (Contributed by Stefan O'Rear, 22-Feb-2015.) |
Ref | Expression |
---|---|
pwnss | ⊢ (𝐴 ∈ 𝑉 → ¬ 𝒫 𝐴 ⊆ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rru 3773 | . . 3 ⊢ ¬ {𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ 𝑥} ∈ 𝐴 | |
2 | ssel 3971 | . . 3 ⊢ (𝒫 𝐴 ⊆ 𝐴 → ({𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ 𝑥} ∈ 𝒫 𝐴 → {𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ 𝑥} ∈ 𝐴)) | |
3 | 1, 2 | mtoi 198 | . 2 ⊢ (𝒫 𝐴 ⊆ 𝐴 → ¬ {𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ 𝑥} ∈ 𝒫 𝐴) |
4 | ssrab2 4073 | . . 3 ⊢ {𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ 𝑥} ⊆ 𝐴 | |
5 | elpw2g 5340 | . . 3 ⊢ (𝐴 ∈ 𝑉 → ({𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ 𝑥} ∈ 𝒫 𝐴 ↔ {𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ 𝑥} ⊆ 𝐴)) | |
6 | 4, 5 | mpbiri 258 | . 2 ⊢ (𝐴 ∈ 𝑉 → {𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ 𝑥} ∈ 𝒫 𝐴) |
7 | 3, 6 | nsyl3 138 | 1 ⊢ (𝐴 ∈ 𝑉 → ¬ 𝒫 𝐴 ⊆ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∈ wcel 2099 {crab 3428 ⊆ wss 3945 𝒫 cpw 4598 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-ext 2699 ax-sep 5293 |
This theorem depends on definitions: df-bi 206 df-an 396 df-tru 1537 df-ex 1775 df-sb 2061 df-clab 2706 df-cleq 2720 df-clel 2806 df-rab 3429 df-v 3472 df-in 3952 df-ss 3962 df-pw 4600 |
This theorem is referenced by: pwne 5346 pwuninel2 8273 |
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