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Theorem rabnc 3791
Description: Law of noncontradiction, in terms of restricted class abstractions. (Contributed by Jeff Madsen, 20-Jun-2011.)
Assertion
Ref Expression
rabnc  |-  ( { x  e.  A  |  ph }  i^i  { x  e.  A  |  -.  ph } )  =  (/)
Distinct variable group:    x, A
Allowed substitution hint:    ph( x)

Proof of Theorem rabnc
StepHypRef Expression
1 inrab 3752 . 2  |-  ( { x  e.  A  |  ph }  i^i  { x  e.  A  |  -.  ph } )  =  {
x  e.  A  | 
( ph  /\  -.  ph ) }
2 rabeq0 3789 . . 3  |-  ( { x  e.  A  | 
( ph  /\  -.  ph ) }  =  (/)  <->  A. x  e.  A  -.  ( ph  /\  -.  ph )
)
3 pm3.24 880 . . . 4  |-  -.  ( ph  /\  -.  ph )
43a1i 11 . . 3  |-  ( x  e.  A  ->  -.  ( ph  /\  -.  ph ) )
52, 4mprgbir 2805 . 2  |-  { x  e.  A  |  ( ph  /\  -.  ph ) }  =  (/)
61, 5eqtri 2470 1  |-  ( { x  e.  A  |  ph }  i^i  { x  e.  A  |  -.  ph } )  =  (/)
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    /\ wa 369    = wceq 1381    e. wcel 1802   {crab 2795    i^i cin 3457   (/)c0 3767
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-ral 2796  df-rex 2797  df-rab 2800  df-v 3095  df-dif 3461  df-in 3465  df-nul 3768
This theorem is referenced by:  hasheuni  27957  ddemeas  28074  ballotth  28342  jm2.22  30905
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