Theorem rabnc 3962

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 3899	. 2 |- ( { x e. A | ph } i^i { x e. A | -. ph } ) = { x e. A | ( ph /\ -. ph ) }
2		pm3.24 926	. . . 4 |- -. ( ph /\ -. ph )
3	2	rgenw 2924	. . 3 |- A. x e. A -. ( ph /\ -. ph )
4		rabeq0 3957	. . 3 |- ( { x e. A | ( ph /\ -. ph ) } = (/) <-> A. x e. A -. ( ph /\ -. ph ) )
5	3, 4	mpbir 221	. 2 |- { x e. A | ( ph /\ -. ph ) } = (/)
6	1, 5	eqtri 2644	1 |- ( { x e. A | ph } i^i { x e. A | -. ph } ) = (/)

Syntax hints:   -. wn 3   /\ wa 384   = wceq 1483   A.wral 2912   {crab 2916   i^i cin 3573   (/)c0 3915

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rab 2921  df-v 3202  df-dif 3577  df-in 3581  df-nul 3916

This theorem is referenced by:  elneldisj  3963  elneldisjOLD  3965  vtxdgoddnumeven  26449  esumrnmpt2  30130  hasheuni  30147  ddemeas  30299  ballotth  30599  jm2.22  37562