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Theorem rabnc 3491
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 3453 . 2  |-  ( { x  e.  A  |  ph }  i^i  { x  e.  A  |  -.  ph } )  =  {
x  e.  A  | 
( ph  /\  -.  ph ) }
2 rabeq0 3489 . . 3  |-  ( { x  e.  A  | 
( ph  /\  -.  ph ) }  =  (/)  <->  A. x  e.  A  -.  ( ph  /\  -.  ph )
)
3 pm3.24 852 . . . 4  |-  -.  ( ph  /\  -.  ph )
43a1i 10 . . 3  |-  ( x  e.  A  ->  -.  ( ph  /\  -.  ph ) )
52, 4mprgbir 2626 . 2  |-  { x  e.  A  |  ( ph  /\  -.  ph ) }  =  (/)
61, 5eqtri 2316 1  |-  ( { x  e.  A  |  ph }  i^i  { x  e.  A  |  -.  ph } )  =  (/)
Colors of variables: wff set class
Syntax hints:   -. wn 3    /\ wa 358    = wceq 1632    e. wcel 1696   {crab 2560    i^i cin 3164   (/)c0 3468
This theorem is referenced by:  ballotth  23112  rabncOLD  26456  jm2.22  27191
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-in 3172  df-nul 3469
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