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Theorem rabnc 3470
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 3422 . 2  |-  ( { x  e.  A  |  ph }  i^i  { x  e.  A  |  -.  ph } )  =  {
x  e.  A  | 
( ph  /\  -.  ph ) }
2 rabeq0 3467 . . 3  |-  ( { x  e.  A  | 
( ph  /\  -.  ph ) }  =  (/)  <->  A. x  e.  A  -.  ( ph  /\  -.  ph )
)
3 pm3.24 694 . . . 4  |-  -.  ( ph  /\  -.  ph )
43a1i 9 . . 3  |-  ( x  e.  A  ->  -.  ( ph  /\  -.  ph ) )
52, 4mprgbir 2548 . 2  |-  { x  e.  A  |  ( ph  /\  -.  ph ) }  =  (/)
61, 5eqtri 2210 1  |-  ( { x  e.  A  |  ph }  i^i  { x  e.  A  |  -.  ph } )  =  (/)
Colors of variables: wff set class
Syntax hints:   -. wn 3    /\ wa 104    = wceq 1364    e. wcel 2160   {crab 2472    i^i cin 3143   (/)c0 3437
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171
This theorem depends on definitions:  df-bi 117  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rab 2477  df-v 2754  df-dif 3146  df-in 3150  df-nul 3438
This theorem is referenced by: (None)
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