Theorem disj5 3890

Description: Two ways of saying that two classes are disjoint. (Contributed by SF, 5-Feb-2015.)

Assertion

Ref	Expression
disj5	|- ((A i^i B) = (/) <-> A C_ ∼ B)

Proof of Theorem disj5

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		vex 2862	. . . . 5 |- x e. _V
2	1	elcompl 3225	. . . 4 |- (x e. ∼ B <-> -. x e. B)
3	2	ralbii 2638	. . 3 |- (A.x e. A x e. ∼ B <-> A.x e. A -. x e. B)
4		df-ral 2619	. . 3 |- (A.x e. A x e. ∼ B <-> A.x(x e. A -> x e. ∼ B))
5	3, 4	bitr3i 242	. 2 |- (A.x e. A -. x e. B <-> A.x(x e. A -> x e. ∼ B))
6		disj 3591	. 2 |- ((A i^i B) = (/) <-> A.x e. A -. x e. B)
7		dfss2 3262	. 2 |- (A C_ ∼ B <-> A.x(x e. A -> x e. ∼ B))
8	5, 6, 7	3bitr4i 268	1 |- ((A i^i B) = (/) <-> A C_ ∼ B)

Syntax hints:  -. wn 3   -> wi 4   <-> wb 176  A.wal 1540   = wceq 1642   e. wcel 1710  A.wral 2614   ∼ ccompl 3205   i^i cin 3208   C_ wss 3257  (/)c0 3550

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 2478  df-ne 2518  df-ral 2619  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-dif 3215  df-ss 3259  df-nul 3551

This theorem is referenced by:  intirr  5029