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Theorem rabnc 3652
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 3614 . 2  |-  ( { x  e.  A  |  ph }  i^i  { x  e.  A  |  -.  ph } )  =  {
x  e.  A  | 
( ph  /\  -.  ph ) }
2 rabeq0 3650 . . 3  |-  ( { x  e.  A  | 
( ph  /\  -.  ph ) }  =  (/)  <->  A. x  e.  A  -.  ( ph  /\  -.  ph )
)
3 pm3.24 854 . . . 4  |-  -.  ( ph  /\  -.  ph )
43a1i 11 . . 3  |-  ( x  e.  A  ->  -.  ( ph  /\  -.  ph ) )
52, 4mprgbir 2777 . 2  |-  { x  e.  A  |  ( ph  /\  -.  ph ) }  =  (/)
61, 5eqtri 2457 1  |-  ( { x  e.  A  |  ph }  i^i  { x  e.  A  |  -.  ph } )  =  (/)
Colors of variables: wff set class
Syntax hints:   -. wn 3    /\ wa 360    = wceq 1653    e. wcel 1726   {crab 2710    i^i cin 3320   (/)c0 3629
This theorem is referenced by:  hasheuni  24476  ballotth  24796  jm2.22  27067
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-ral 2711  df-rex 2712  df-rab 2715  df-v 2959  df-dif 3324  df-in 3328  df-nul 3630
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