Theorem rabnc 3575

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 3528	. 2 |- ({x e. A | ph} i^i {x e. A | -. ph}) = {x e. A | (ph /\ -. ph)}
2		rabeq0 3573	. . 3 |- ({x e. A | (ph /\ -. ph)} = (/) <-> A.x e. A -. (ph /\ -. ph))
3		pm3.24 852	. . . 4 |- -. (ph /\ -. ph)
4	3	a1i 10	. . 3 |- (x e. A -> -. (ph /\ -. ph))
5	2, 4	mprgbir 2685	. 2 |- {x e. A | (ph /\ -. ph)} = (/)
6	1, 5	eqtri 2373	1 |- ({x e. A | ph} i^i {x e. A | -. ph}) = (/)

Syntax hints:  -. wn 3   /\ wa 358   = wceq 1642   e. wcel 1710  {crab 2619   i^i cin 3209  (/)c0 3551

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ne 2519  df-ral 2620  df-rex 2621  df-rab 2624  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-dif 3216  df-nul 3552

This theorem is referenced by: (None)