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Theorem rabnc 3395
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 3348 . 2  |-  ( { x  e.  A  |  ph }  i^i  { x  e.  A  |  -.  ph } )  =  {
x  e.  A  | 
( ph  /\  -.  ph ) }
2 rabeq0 3392 . . 3  |-  ( { x  e.  A  | 
( ph  /\  -.  ph ) }  =  (/)  <->  A. x  e.  A  -.  ( ph  /\  -.  ph )
)
3 pm3.24 682 . . . 4  |-  -.  ( ph  /\  -.  ph )
43a1i 9 . . 3  |-  ( x  e.  A  ->  -.  ( ph  /\  -.  ph ) )
52, 4mprgbir 2490 . 2  |-  { x  e.  A  |  ( ph  /\  -.  ph ) }  =  (/)
61, 5eqtri 2160 1  |-  ( { x  e.  A  |  ph }  i^i  { x  e.  A  |  -.  ph } )  =  (/)
Colors of variables: wff set class
Syntax hints:   -. wn 3    /\ wa 103    = wceq 1331    e. wcel 1480   {crab 2420    i^i cin 3070   (/)c0 3363
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121
This theorem depends on definitions:  df-bi 116  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rab 2425  df-v 2688  df-dif 3073  df-in 3077  df-nul 3364
This theorem is referenced by: (None)
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