Theorem rabnc 3453

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 3405	. 2 |- ( { x e. A | ph } i^i { x e. A | -. ph } ) = { x e. A | ( ph /\ -. ph ) }
2		rabeq0 3450	. . 3 |- ( { x e. A | ( ph /\ -. ph ) } = (/) <-> A. x e. A -. ( ph /\ -. ph ) )
3		pm3.24 693	. . . 4 |- -. ( ph /\ -. ph )
4	3	a1i 9	. . 3 |- ( x e. A -> -. ( ph /\ -. ph ) )
5	2, 4	mprgbir 2533	. 2 |- { x e. A | ( ph /\ -. ph ) } = (/)
6	1, 5	eqtri 2196	1 |- ( { x e. A | ph } i^i { x e. A | -. ph } ) = (/)

Syntax hints:   -. wn 3   /\ wa 104   = wceq 1353   e. wcel 2146   {crab 2457   i^i cin 3126   (/)c0 3420

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 614  ax-in2 615  ax-io 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-ext 2157

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-fal 1359  df-nf 1459  df-sb 1761  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ral 2458  df-rab 2462  df-v 2737  df-dif 3129  df-in 3133  df-nul 3421

This theorem is referenced by: (None)