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Theorem pwnss 4860
Description: The power set of a set is never a subset. (Contributed by Stefan O'Rear, 22-Feb-2015.)
Assertion
Ref Expression
pwnss (𝐴𝑉 → ¬ 𝒫 𝐴𝐴)

Proof of Theorem pwnss
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eleq12 2720 . . . . . . 7 ((𝑦 = {𝑥𝐴𝑥𝑥} ∧ 𝑦 = {𝑥𝐴𝑥𝑥}) → (𝑦𝑦 ↔ {𝑥𝐴𝑥𝑥} ∈ {𝑥𝐴𝑥𝑥}))
21anidms 678 . . . . . 6 (𝑦 = {𝑥𝐴𝑥𝑥} → (𝑦𝑦 ↔ {𝑥𝐴𝑥𝑥} ∈ {𝑥𝐴𝑥𝑥}))
32notbid 307 . . . . 5 (𝑦 = {𝑥𝐴𝑥𝑥} → (¬ 𝑦𝑦 ↔ ¬ {𝑥𝐴𝑥𝑥} ∈ {𝑥𝐴𝑥𝑥}))
4 df-nel 2927 . . . . . . 7 (𝑥𝑥 ↔ ¬ 𝑥𝑥)
5 eleq12 2720 . . . . . . . . 9 ((𝑥 = 𝑦𝑥 = 𝑦) → (𝑥𝑥𝑦𝑦))
65anidms 678 . . . . . . . 8 (𝑥 = 𝑦 → (𝑥𝑥𝑦𝑦))
76notbid 307 . . . . . . 7 (𝑥 = 𝑦 → (¬ 𝑥𝑥 ↔ ¬ 𝑦𝑦))
84, 7syl5bb 272 . . . . . 6 (𝑥 = 𝑦 → (𝑥𝑥 ↔ ¬ 𝑦𝑦))
98cbvrabv 3230 . . . . 5 {𝑥𝐴𝑥𝑥} = {𝑦𝐴 ∣ ¬ 𝑦𝑦}
103, 9elrab2 3399 . . . 4 ({𝑥𝐴𝑥𝑥} ∈ {𝑥𝐴𝑥𝑥} ↔ ({𝑥𝐴𝑥𝑥} ∈ 𝐴 ∧ ¬ {𝑥𝐴𝑥𝑥} ∈ {𝑥𝐴𝑥𝑥}))
11 pclem6 991 . . . 4 (({𝑥𝐴𝑥𝑥} ∈ {𝑥𝐴𝑥𝑥} ↔ ({𝑥𝐴𝑥𝑥} ∈ 𝐴 ∧ ¬ {𝑥𝐴𝑥𝑥} ∈ {𝑥𝐴𝑥𝑥})) → ¬ {𝑥𝐴𝑥𝑥} ∈ 𝐴)
1210, 11ax-mp 5 . . 3 ¬ {𝑥𝐴𝑥𝑥} ∈ 𝐴
13 ssel 3630 . . 3 (𝒫 𝐴𝐴 → ({𝑥𝐴𝑥𝑥} ∈ 𝒫 𝐴 → {𝑥𝐴𝑥𝑥} ∈ 𝐴))
1412, 13mtoi 190 . 2 (𝒫 𝐴𝐴 → ¬ {𝑥𝐴𝑥𝑥} ∈ 𝒫 𝐴)
15 ssrab2 3720 . . 3 {𝑥𝐴𝑥𝑥} ⊆ 𝐴
16 elpw2g 4857 . . 3 (𝐴𝑉 → ({𝑥𝐴𝑥𝑥} ∈ 𝒫 𝐴 ↔ {𝑥𝐴𝑥𝑥} ⊆ 𝐴))
1715, 16mpbiri 248 . 2 (𝐴𝑉 → {𝑥𝐴𝑥𝑥} ∈ 𝒫 𝐴)
1814, 17nsyl3 133 1 (𝐴𝑉 → ¬ 𝒫 𝐴𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 383   = wceq 1523  wcel 2030  wnel 2926  {crab 2945  wss 3607  𝒫 cpw 4191
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-nel 2927  df-rab 2950  df-v 3233  df-in 3614  df-ss 3621  df-pw 4193
This theorem is referenced by:  pwne  4861  pwuninel2  7445
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