MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  2pwne Structured version   Unicode version

Theorem 2pwne 7482
Description: No set equals the power set of its power set. (Contributed by NM, 17-Nov-2008.)
Assertion
Ref Expression
2pwne  |-  ( A  e.  V  ->  ~P ~P A  =/=  A
)

Proof of Theorem 2pwne
StepHypRef Expression
1 sdomirr 7463 . . 3  |-  -.  ~P ~P A  ~<  ~P ~P A
2 canth2g 7480 . . . . 5  |-  ( A  e.  V  ->  A  ~<  ~P A )
3 pwexg 4491 . . . . . 6  |-  ( A  e.  V  ->  ~P A  e.  _V )
4 canth2g 7480 . . . . . 6  |-  ( ~P A  e.  _V  ->  ~P A  ~<  ~P ~P A )
53, 4syl 16 . . . . 5  |-  ( A  e.  V  ->  ~P A  ~<  ~P ~P A
)
6 sdomtr 7464 . . . . 5  |-  ( ( A  ~<  ~P A  /\  ~P A  ~<  ~P ~P A )  ->  A  ~<  ~P ~P A )
72, 5, 6syl2anc 661 . . . 4  |-  ( A  e.  V  ->  A  ~<  ~P ~P A )
8 breq1 4310 . . . 4  |-  ( ~P ~P A  =  A  ->  ( ~P ~P A  ~<  ~P ~P A  <->  A 
~<  ~P ~P A ) )
97, 8syl5ibrcom 222 . . 3  |-  ( A  e.  V  ->  ( ~P ~P A  =  A  ->  ~P ~P A  ~<  ~P ~P A ) )
101, 9mtoi 178 . 2  |-  ( A  e.  V  ->  -.  ~P ~P A  =  A )
1110neneqad 2696 1  |-  ( A  e.  V  ->  ~P ~P A  =/=  A
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1369    e. wcel 1756    =/= wne 2620   _Vcvv 2987   ~Pcpw 3875   class class class wbr 4307    ~< csdm 7324
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4428  ax-nul 4436  ax-pow 4485  ax-pr 4546  ax-un 6387
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-ral 2735  df-rex 2736  df-rab 2739  df-v 2989  df-sbc 3202  df-csb 3304  df-dif 3346  df-un 3348  df-in 3350  df-ss 3357  df-nul 3653  df-if 3807  df-pw 3877  df-sn 3893  df-pr 3895  df-op 3899  df-uni 4107  df-br 4308  df-opab 4366  df-mpt 4367  df-id 4651  df-xp 4861  df-rel 4862  df-cnv 4863  df-co 4864  df-dm 4865  df-rn 4866  df-res 4867  df-ima 4868  df-iota 5396  df-fun 5435  df-fn 5436  df-f 5437  df-f1 5438  df-fo 5439  df-f1o 5440  df-fv 5441  df-er 7116  df-en 7326  df-dom 7327  df-sdom 7328
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator