Theorem reldisj 3502

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 3172	. . . 4 |- ( A C_ C <-> A. x ( x e. A -> x e. C ) )
2		pm5.44 926	. . . . . 6 |- ( ( x e. A -> x e. C ) -> ( ( x e. A -> -. x e. B ) <-> ( x e. A -> ( x e. C /\ -. x e. B ) ) ) )
3		eldif 3166	. . . . . . 7 |- ( x e. ( C \ B ) <-> ( x e. C /\ -. x e. B ) )
4	3	imbi2i 226	. . . . . 6 |- ( ( x e. A -> x e. ( C \ B ) ) <-> ( x e. A -> ( x e. C /\ -. x e. B ) ) )
5	2, 4	bitr4di 198	. . . . 5 |- ( ( x e. A -> x e. C ) -> ( ( x e. A -> -. x e. B ) <-> ( x e. A -> x e. ( C \ B ) ) ) )
6	5	sps 1551	. . . 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 121	. . 3 |- ( A C_ C -> ( ( x e. A -> -. x e. B ) <-> ( x e. A -> x e. ( C \ B ) ) ) )
8	7	albidv 1838	. 2 |- ( A C_ C -> ( A. x ( x e. A -> -. x e. B ) <-> A. x ( x e. A -> x e. ( C \ B ) ) ) )
9		disj1 3501	. 2 |- ( ( A i^i B ) = (/) <-> A. x ( x e. A -> -. x e. B ) )
10		dfss2 3172	. 2 |- ( A C_ ( C \ B ) <-> A. x ( x e. A -> x e. ( C \ B ) ) )
11	8, 9, 10	3bitr4g 223	1 |- ( A C_ C -> ( ( A i^i B ) = (/) <-> A C_ ( C \ B ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   A.wal 1362   = wceq 1364   e. wcel 2167   \ cdif 3154   i^i cin 3156   C_ wss 3157   (/)c0 3450

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-v 2765  df-dif 3159  df-in 3163  df-ss 3170  df-nul 3451

This theorem is referenced by:  disj2  3506  ssdifsn  3750  structcnvcnv  12694