Theorem rabnc 3390

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

Step	Hyp	Ref	Expression
1		inrab 3343	. 2 |- ( { x e. A | ph } i^i { x e. A | -. ph } ) = { x e. A | ( ph /\ -. ph ) }
2		rabeq0 3387	. . 3 |- ( { x e. A | ( ph /\ -. ph ) } = (/) <-> A. x e. A -. ( ph /\ -. ph ) )
3		pm3.24 682	. . . 4 |- -. ( ph /\ -. ph )
4	3	a1i 9	. . 3 |- ( x e. A -> -. ( ph /\ -. ph ) )
5	2, 4	mprgbir 2488	. 2 |- { x e. A | ( ph /\ -. ph ) } = (/)
6	1, 5	eqtri 2158	1 |- ( { x e. A | ph } i^i { x e. A | -. ph } ) = (/)

Syntax hints:   -. wn 3   /\ wa 103   = wceq 1331   e. wcel 1480   {crab 2418   i^i cin 3065   (/)c0 3358

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 2119

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rab 2423  df-v 2683  df-dif 3068  df-in 3072  df-nul 3359

This theorem is referenced by: (None)