MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  pwuninel2 Structured version   Visualization version   GIF version

Theorem pwuninel2 7567
Description: Direct proof of pwuninel 7568 avoiding functions and thus several ZF axioms. (Contributed by Stefan O'Rear, 22-Feb-2015.)
Assertion
Ref Expression
pwuninel2 ( 𝐴𝑉 → ¬ 𝒫 𝐴𝐴)

Proof of Theorem pwuninel2
StepHypRef Expression
1 pwnss 4977 . 2 ( 𝐴𝑉 → ¬ 𝒫 𝐴 𝐴)
2 elssuni 4617 . 2 (𝒫 𝐴𝐴 → 𝒫 𝐴 𝐴)
31, 2nsyl 135 1 ( 𝐴𝑉 → ¬ 𝒫 𝐴𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wcel 2137  wss 3713  𝒫 cpw 4300   cuni 4586
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1986  ax-6 2052  ax-7 2088  ax-9 2146  ax-10 2166  ax-11 2181  ax-12 2194  ax-13 2389  ax-ext 2738  ax-sep 4931
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2045  df-clab 2745  df-cleq 2751  df-clel 2754  df-nfc 2889  df-nel 3034  df-rab 3057  df-v 3340  df-in 3720  df-ss 3727  df-pw 4302  df-uni 4587
This theorem is referenced by:  pwuninel  7568
  Copyright terms: Public domain W3C validator