Theorem rabnc 3501

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 3453	. 2 |- ( { x e. A | ph } i^i { x e. A | -. ph } ) = { x e. A | ( ph /\ -. ph ) }
2		rabeq0 3498	. . 3 |- ( { x e. A | ( ph /\ -. ph ) } = (/) <-> A. x e. A -. ( ph /\ -. ph ) )
3		pm3.24 695	. . . 4 |- -. ( ph /\ -. ph )
4	3	a1i 9	. . 3 |- ( x e. A -> -. ( ph /\ -. ph ) )
5	2, 4	mprgbir 2566	. 2 |- { x e. A | ( ph /\ -. ph ) } = (/)
6	1, 5	eqtri 2228	1 |- ( { x e. A | ph } i^i { x e. A | -. ph } ) = (/)

Syntax hints:   -. wn 3   /\ wa 104   = wceq 1373   e. wcel 2178   {crab 2490   i^i cin 3173   (/)c0 3468

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 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ral 2491  df-rab 2495  df-v 2778  df-dif 3176  df-in 3180  df-nul 3469

This theorem is referenced by: (None)