Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  pwnss Structured version   GIF version

Theorem pwnss 3903
 Description: The power set of a set is never a subset. (Contributed by Stefan O'Rear, 22-Feb-2015.)
Assertion
Ref Expression
pwnss (A 𝑉 → ¬ 𝒫 AA)

Proof of Theorem pwnss
Dummy variables x y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eleq12 2099 . . . . . . 7 ((y = {x Axx} y = {x Axx}) → (y y ↔ {x Axx} {x Axx}))
21anidms 377 . . . . . 6 (y = {x Axx} → (y y ↔ {x Axx} {x Axx}))
32notbid 591 . . . . 5 (y = {x Axx} → (¬ y y ↔ ¬ {x Axx} {x Axx}))
4 df-nel 2204 . . . . . . 7 (xx ↔ ¬ x x)
5 eleq12 2099 . . . . . . . . 9 ((x = y x = y) → (x xy y))
65anidms 377 . . . . . . . 8 (x = y → (x xy y))
76notbid 591 . . . . . . 7 (x = y → (¬ x x ↔ ¬ y y))
84, 7syl5bb 181 . . . . . 6 (x = y → (xx ↔ ¬ y y))
98cbvrabv 2550 . . . . 5 {x Axx} = {y A ∣ ¬ y y}
103, 9elrab2 2694 . . . 4 ({x Axx} {x Axx} ↔ ({x Axx} A ¬ {x Axx} {x Axx}))
11 pclem6 1264 . . . 4 (({x Axx} {x Axx} ↔ ({x Axx} A ¬ {x Axx} {x Axx})) → ¬ {x Axx} A)
1210, 11ax-mp 7 . . 3 ¬ {x Axx} A
13 ssel 2933 . . 3 (𝒫 AA → ({x Axx} 𝒫 A → {x Axx} A))
1412, 13mtoi 589 . 2 (𝒫 AA → ¬ {x Axx} 𝒫 A)
15 ssrab2 3019 . . 3 {x Axx} ⊆ A
16 elpw2g 3901 . . 3 (A 𝑉 → ({x Axx} 𝒫 A ↔ {x Axx} ⊆ A))
1715, 16mpbiri 157 . 2 (A 𝑉 → {x Axx} 𝒫 A)
1814, 17nsyl3 556 1 (A 𝑉 → ¬ 𝒫 AA)
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 97   ↔ wb 98   = wceq 1242   ∈ wcel 1390   ∉ wnel 2202  {crab 2304   ⊆ wss 2911  𝒫 cpw 3351 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866 This theorem depends on definitions:  df-bi 110  df-tru 1245  df-fal 1248  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-nel 2204  df-rab 2309  df-v 2553  df-in 2918  df-ss 2925  df-pw 3353 This theorem is referenced by:  pwne  3904  pwuninel2  5838
 Copyright terms: Public domain W3C validator