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Theorem pwnss 3986
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 2152 . . . . . . 7 ((𝑦 = {𝑥𝐴𝑥𝑥} ∧ 𝑦 = {𝑥𝐴𝑥𝑥}) → (𝑦𝑦 ↔ {𝑥𝐴𝑥𝑥} ∈ {𝑥𝐴𝑥𝑥}))
21anidms 389 . . . . . 6 (𝑦 = {𝑥𝐴𝑥𝑥} → (𝑦𝑦 ↔ {𝑥𝐴𝑥𝑥} ∈ {𝑥𝐴𝑥𝑥}))
32notbid 627 . . . . 5 (𝑦 = {𝑥𝐴𝑥𝑥} → (¬ 𝑦𝑦 ↔ ¬ {𝑥𝐴𝑥𝑥} ∈ {𝑥𝐴𝑥𝑥}))
4 df-nel 2351 . . . . . . 7 (𝑥𝑥 ↔ ¬ 𝑥𝑥)
5 eleq12 2152 . . . . . . . . 9 ((𝑥 = 𝑦𝑥 = 𝑦) → (𝑥𝑥𝑦𝑦))
65anidms 389 . . . . . . . 8 (𝑥 = 𝑦 → (𝑥𝑥𝑦𝑦))
76notbid 627 . . . . . . 7 (𝑥 = 𝑦 → (¬ 𝑥𝑥 ↔ ¬ 𝑦𝑦))
84, 7syl5bb 190 . . . . . 6 (𝑥 = 𝑦 → (𝑥𝑥 ↔ ¬ 𝑦𝑦))
98cbvrabv 2618 . . . . 5 {𝑥𝐴𝑥𝑥} = {𝑦𝐴 ∣ ¬ 𝑦𝑦}
103, 9elrab2 2772 . . . 4 ({𝑥𝐴𝑥𝑥} ∈ {𝑥𝐴𝑥𝑥} ↔ ({𝑥𝐴𝑥𝑥} ∈ 𝐴 ∧ ¬ {𝑥𝐴𝑥𝑥} ∈ {𝑥𝐴𝑥𝑥}))
11 pclem6 1310 . . . 4 (({𝑥𝐴𝑥𝑥} ∈ {𝑥𝐴𝑥𝑥} ↔ ({𝑥𝐴𝑥𝑥} ∈ 𝐴 ∧ ¬ {𝑥𝐴𝑥𝑥} ∈ {𝑥𝐴𝑥𝑥})) → ¬ {𝑥𝐴𝑥𝑥} ∈ 𝐴)
1210, 11ax-mp 7 . . 3 ¬ {𝑥𝐴𝑥𝑥} ∈ 𝐴
13 ssel 3017 . . 3 (𝒫 𝐴𝐴 → ({𝑥𝐴𝑥𝑥} ∈ 𝒫 𝐴 → {𝑥𝐴𝑥𝑥} ∈ 𝐴))
1412, 13mtoi 625 . 2 (𝒫 𝐴𝐴 → ¬ {𝑥𝐴𝑥𝑥} ∈ 𝒫 𝐴)
15 ssrab2 3104 . . 3 {𝑥𝐴𝑥𝑥} ⊆ 𝐴
16 elpw2g 3984 . . 3 (𝐴𝑉 → ({𝑥𝐴𝑥𝑥} ∈ 𝒫 𝐴 ↔ {𝑥𝐴𝑥𝑥} ⊆ 𝐴))
1715, 16mpbiri 166 . 2 (𝐴𝑉 → {𝑥𝐴𝑥𝑥} ∈ 𝒫 𝐴)
1814, 17nsyl3 591 1 (𝐴𝑉 → ¬ 𝒫 𝐴𝐴)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 102  wb 103   = wceq 1289  wcel 1438  wnel 2350  {crab 2363  wss 2997  𝒫 cpw 3425
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-nel 2351  df-rab 2368  df-v 2621  df-in 3003  df-ss 3010  df-pw 3427
This theorem is referenced by:  pwne  3987  pwuninel2  6029
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