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Theorem pwnss 4053
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 2182 . . . . . . 7 ((𝑦 = {𝑥𝐴𝑥𝑥} ∧ 𝑦 = {𝑥𝐴𝑥𝑥}) → (𝑦𝑦 ↔ {𝑥𝐴𝑥𝑥} ∈ {𝑥𝐴𝑥𝑥}))
21anidms 394 . . . . . 6 (𝑦 = {𝑥𝐴𝑥𝑥} → (𝑦𝑦 ↔ {𝑥𝐴𝑥𝑥} ∈ {𝑥𝐴𝑥𝑥}))
32notbid 641 . . . . 5 (𝑦 = {𝑥𝐴𝑥𝑥} → (¬ 𝑦𝑦 ↔ ¬ {𝑥𝐴𝑥𝑥} ∈ {𝑥𝐴𝑥𝑥}))
4 df-nel 2381 . . . . . . 7 (𝑥𝑥 ↔ ¬ 𝑥𝑥)
5 eleq12 2182 . . . . . . . . 9 ((𝑥 = 𝑦𝑥 = 𝑦) → (𝑥𝑥𝑦𝑦))
65anidms 394 . . . . . . . 8 (𝑥 = 𝑦 → (𝑥𝑥𝑦𝑦))
76notbid 641 . . . . . . 7 (𝑥 = 𝑦 → (¬ 𝑥𝑥 ↔ ¬ 𝑦𝑦))
84, 7syl5bb 191 . . . . . 6 (𝑥 = 𝑦 → (𝑥𝑥 ↔ ¬ 𝑦𝑦))
98cbvrabv 2659 . . . . 5 {𝑥𝐴𝑥𝑥} = {𝑦𝐴 ∣ ¬ 𝑦𝑦}
103, 9elrab2 2816 . . . 4 ({𝑥𝐴𝑥𝑥} ∈ {𝑥𝐴𝑥𝑥} ↔ ({𝑥𝐴𝑥𝑥} ∈ 𝐴 ∧ ¬ {𝑥𝐴𝑥𝑥} ∈ {𝑥𝐴𝑥𝑥}))
11 pclem6 1337 . . . 4 (({𝑥𝐴𝑥𝑥} ∈ {𝑥𝐴𝑥𝑥} ↔ ({𝑥𝐴𝑥𝑥} ∈ 𝐴 ∧ ¬ {𝑥𝐴𝑥𝑥} ∈ {𝑥𝐴𝑥𝑥})) → ¬ {𝑥𝐴𝑥𝑥} ∈ 𝐴)
1210, 11ax-mp 5 . . 3 ¬ {𝑥𝐴𝑥𝑥} ∈ 𝐴
13 ssel 3061 . . 3 (𝒫 𝐴𝐴 → ({𝑥𝐴𝑥𝑥} ∈ 𝒫 𝐴 → {𝑥𝐴𝑥𝑥} ∈ 𝐴))
1412, 13mtoi 638 . 2 (𝒫 𝐴𝐴 → ¬ {𝑥𝐴𝑥𝑥} ∈ 𝒫 𝐴)
15 ssrab2 3152 . . 3 {𝑥𝐴𝑥𝑥} ⊆ 𝐴
16 elpw2g 4051 . . 3 (𝐴𝑉 → ({𝑥𝐴𝑥𝑥} ∈ 𝒫 𝐴 ↔ {𝑥𝐴𝑥𝑥} ⊆ 𝐴))
1715, 16mpbiri 167 . 2 (𝐴𝑉 → {𝑥𝐴𝑥𝑥} ∈ 𝒫 𝐴)
1814, 17nsyl3 600 1 (𝐴𝑉 → ¬ 𝒫 𝐴𝐴)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 103  wb 104   = wceq 1316  wcel 1465  wnel 2380  {crab 2397  wss 3041  𝒫 cpw 3480
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 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016
This theorem depends on definitions:  df-bi 116  df-tru 1319  df-fal 1322  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-nel 2381  df-rab 2402  df-v 2662  df-in 3047  df-ss 3054  df-pw 3482
This theorem is referenced by:  pwne  4054  pwuninel2  6147
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