Theorem reldisj 3466

Description: Two ways of saying that two classes are disjoint, using the complement of B relative to a universe C. (Contributed by NM, 15-Feb-2007.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)

Assertion

Ref	Expression
reldisj	|- ( A C_ C -> ( ( A i^i B ) = (/) <-> A C_ ( C \ B ) ) )

Proof of Theorem reldisj

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfss2 3136	. . . 4 |- ( A C_ C <-> A. x ( x e. A -> x e. C ) )
2		pm5.44 920	. . . . . 6 |- ( ( x e. A -> x e. C ) -> ( ( x e. A -> -. x e. B ) <-> ( x e. A -> ( x e. C /\ -. x e. B ) ) ) )
3		eldif 3130	. . . . . . 7 |- ( x e. ( C \ B ) <-> ( x e. C /\ -. x e. B ) )
4	3	imbi2i 225	. . . . . 6 |- ( ( x e. A -> x e. ( C \ B ) ) <-> ( x e. A -> ( x e. C /\ -. x e. B ) ) )
5	2, 4	bitr4di 197	. . . . 5 |- ( ( x e. A -> x e. C ) -> ( ( x e. A -> -. x e. B ) <-> ( x e. A -> x e. ( C \ B ) ) ) )
6	5	sps 1530	. . . 4 |- ( A. x ( x e. A -> x e. C ) -> ( ( x e. A -> -. x e. B ) <-> ( x e. A -> x e. ( C \ B ) ) ) )
7	1, 6	sylbi 120	. . 3 |- ( A C_ C -> ( ( x e. A -> -. x e. B ) <-> ( x e. A -> x e. ( C \ B ) ) ) )
8	7	albidv 1817	. 2 |- ( A C_ C -> ( A. x ( x e. A -> -. x e. B ) <-> A. x ( x e. A -> x e. ( C \ B ) ) ) )
9		disj1 3465	. 2 |- ( ( A i^i B ) = (/) <-> A. x ( x e. A -> -. x e. B ) )
10		dfss2 3136	. 2 |- ( A C_ ( C \ B ) <-> A. x ( x e. A -> x e. ( C \ B ) ) )
11	8, 9, 10	3bitr4g 222	1 |- ( A C_ C -> ( ( A i^i B ) = (/) <-> A C_ ( C \ B ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   <-> wb 104   A.wal 1346   = wceq 1348   e. wcel 2141   \ cdif 3118   i^i cin 3120   C_ wss 3121   (/)c0 3414

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-v 2732  df-dif 3123  df-in 3127  df-ss 3134  df-nul 3415

This theorem is referenced by:  disj2  3470  ssdifsn  3711  structcnvcnv  12432