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Theorem pwnss 4795
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 2694 . . . . . . 7 ((𝑦 = {𝑥𝐴𝑥𝑥} ∧ 𝑦 = {𝑥𝐴𝑥𝑥}) → (𝑦𝑦 ↔ {𝑥𝐴𝑥𝑥} ∈ {𝑥𝐴𝑥𝑥}))
21anidms 676 . . . . . 6 (𝑦 = {𝑥𝐴𝑥𝑥} → (𝑦𝑦 ↔ {𝑥𝐴𝑥𝑥} ∈ {𝑥𝐴𝑥𝑥}))
32notbid 308 . . . . 5 (𝑦 = {𝑥𝐴𝑥𝑥} → (¬ 𝑦𝑦 ↔ ¬ {𝑥𝐴𝑥𝑥} ∈ {𝑥𝐴𝑥𝑥}))
4 df-nel 2900 . . . . . . 7 (𝑥𝑥 ↔ ¬ 𝑥𝑥)
5 eleq12 2694 . . . . . . . . 9 ((𝑥 = 𝑦𝑥 = 𝑦) → (𝑥𝑥𝑦𝑦))
65anidms 676 . . . . . . . 8 (𝑥 = 𝑦 → (𝑥𝑥𝑦𝑦))
76notbid 308 . . . . . . 7 (𝑥 = 𝑦 → (¬ 𝑥𝑥 ↔ ¬ 𝑦𝑦))
84, 7syl5bb 272 . . . . . 6 (𝑥 = 𝑦 → (𝑥𝑥 ↔ ¬ 𝑦𝑦))
98cbvrabv 3190 . . . . 5 {𝑥𝐴𝑥𝑥} = {𝑦𝐴 ∣ ¬ 𝑦𝑦}
103, 9elrab2 3353 . . . 4 ({𝑥𝐴𝑥𝑥} ∈ {𝑥𝐴𝑥𝑥} ↔ ({𝑥𝐴𝑥𝑥} ∈ 𝐴 ∧ ¬ {𝑥𝐴𝑥𝑥} ∈ {𝑥𝐴𝑥𝑥}))
11 pclem6 970 . . . 4 (({𝑥𝐴𝑥𝑥} ∈ {𝑥𝐴𝑥𝑥} ↔ ({𝑥𝐴𝑥𝑥} ∈ 𝐴 ∧ ¬ {𝑥𝐴𝑥𝑥} ∈ {𝑥𝐴𝑥𝑥})) → ¬ {𝑥𝐴𝑥𝑥} ∈ 𝐴)
1210, 11ax-mp 5 . . 3 ¬ {𝑥𝐴𝑥𝑥} ∈ 𝐴
13 ssel 3582 . . 3 (𝒫 𝐴𝐴 → ({𝑥𝐴𝑥𝑥} ∈ 𝒫 𝐴 → {𝑥𝐴𝑥𝑥} ∈ 𝐴))
1412, 13mtoi 190 . 2 (𝒫 𝐴𝐴 → ¬ {𝑥𝐴𝑥𝑥} ∈ 𝒫 𝐴)
15 ssrab2 3671 . . 3 {𝑥𝐴𝑥𝑥} ⊆ 𝐴
16 elpw2g 4792 . . 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 384   = wceq 1480  wcel 1992  wnel 2899  {crab 2916  wss 3560  𝒫 cpw 4135
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-sep 4746
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-nel 2900  df-rab 2921  df-v 3193  df-in 3567  df-ss 3574  df-pw 4137
This theorem is referenced by:  pwne  4796  pwuninel2  7346
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