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Theorem pwnss 4143
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 2235 . . . . . . 7 ((𝑦 = {𝑥𝐴𝑥𝑥} ∧ 𝑦 = {𝑥𝐴𝑥𝑥}) → (𝑦𝑦 ↔ {𝑥𝐴𝑥𝑥} ∈ {𝑥𝐴𝑥𝑥}))
21anidms 395 . . . . . 6 (𝑦 = {𝑥𝐴𝑥𝑥} → (𝑦𝑦 ↔ {𝑥𝐴𝑥𝑥} ∈ {𝑥𝐴𝑥𝑥}))
32notbid 662 . . . . 5 (𝑦 = {𝑥𝐴𝑥𝑥} → (¬ 𝑦𝑦 ↔ ¬ {𝑥𝐴𝑥𝑥} ∈ {𝑥𝐴𝑥𝑥}))
4 df-nel 2436 . . . . . . 7 (𝑥𝑥 ↔ ¬ 𝑥𝑥)
5 eleq12 2235 . . . . . . . . 9 ((𝑥 = 𝑦𝑥 = 𝑦) → (𝑥𝑥𝑦𝑦))
65anidms 395 . . . . . . . 8 (𝑥 = 𝑦 → (𝑥𝑥𝑦𝑦))
76notbid 662 . . . . . . 7 (𝑥 = 𝑦 → (¬ 𝑥𝑥 ↔ ¬ 𝑦𝑦))
84, 7syl5bb 191 . . . . . 6 (𝑥 = 𝑦 → (𝑥𝑥 ↔ ¬ 𝑦𝑦))
98cbvrabv 2729 . . . . 5 {𝑥𝐴𝑥𝑥} = {𝑦𝐴 ∣ ¬ 𝑦𝑦}
103, 9elrab2 2889 . . . 4 ({𝑥𝐴𝑥𝑥} ∈ {𝑥𝐴𝑥𝑥} ↔ ({𝑥𝐴𝑥𝑥} ∈ 𝐴 ∧ ¬ {𝑥𝐴𝑥𝑥} ∈ {𝑥𝐴𝑥𝑥}))
11 pclem6 1369 . . . 4 (({𝑥𝐴𝑥𝑥} ∈ {𝑥𝐴𝑥𝑥} ↔ ({𝑥𝐴𝑥𝑥} ∈ 𝐴 ∧ ¬ {𝑥𝐴𝑥𝑥} ∈ {𝑥𝐴𝑥𝑥})) → ¬ {𝑥𝐴𝑥𝑥} ∈ 𝐴)
1210, 11ax-mp 5 . . 3 ¬ {𝑥𝐴𝑥𝑥} ∈ 𝐴
13 ssel 3141 . . 3 (𝒫 𝐴𝐴 → ({𝑥𝐴𝑥𝑥} ∈ 𝒫 𝐴 → {𝑥𝐴𝑥𝑥} ∈ 𝐴))
1412, 13mtoi 659 . 2 (𝒫 𝐴𝐴 → ¬ {𝑥𝐴𝑥𝑥} ∈ 𝒫 𝐴)
15 ssrab2 3232 . . 3 {𝑥𝐴𝑥𝑥} ⊆ 𝐴
16 elpw2g 4140 . . 3 (𝐴𝑉 → ({𝑥𝐴𝑥𝑥} ∈ 𝒫 𝐴 ↔ {𝑥𝐴𝑥𝑥} ⊆ 𝐴))
1715, 16mpbiri 167 . 2 (𝐴𝑉 → {𝑥𝐴𝑥𝑥} ∈ 𝒫 𝐴)
1814, 17nsyl3 621 1 (𝐴𝑉 → ¬ 𝒫 𝐴𝐴)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 103  wb 104   = wceq 1348  wcel 2141  wnel 2435  {crab 2452  wss 3121  𝒫 cpw 3564
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152  ax-sep 4105
This theorem depends on definitions:  df-bi 116  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-nel 2436  df-rab 2457  df-v 2732  df-in 3127  df-ss 3134  df-pw 3566
This theorem is referenced by:  pwne  4144  pwuninel2  6259
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