Theorem rabnc 3524

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 3476	. 2 |- ( { x e. A | ph } i^i { x e. A | -. ph } ) = { x e. A | ( ph /\ -. ph ) }
2		rabeq0 3521	. . 3 |- ( { x e. A | ( ph /\ -. ph ) } = (/) <-> A. x e. A -. ( ph /\ -. ph ) )
3		pm3.24 698	. . . 4 |- -. ( ph /\ -. ph )
4	3	a1i 9	. . 3 |- ( x e. A -> -. ( ph /\ -. ph ) )
5	2, 4	mprgbir 2588	. 2 |- { x e. A | ( ph /\ -. ph ) } = (/)
6	1, 5	eqtri 2250	1 |- ( { x e. A | ph } i^i { x e. A | -. ph } ) = (/)

Syntax hints:   -. wn 3   /\ wa 104   = wceq 1395   e. wcel 2200   {crab 2512   i^i cin 3196   (/)c0 3491

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rab 2517  df-v 2801  df-dif 3199  df-in 3203  df-nul 3492

This theorem is referenced by: (None)