Theorem reldisj 3311

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 2997	. . . 4 |- ( A C_ C <-> A. x ( x e. A -> x e. C ) )
2		pm5.44 868	. . . . . 6 |- ( ( x e. A -> x e. C ) -> ( ( x e. A -> -. x e. B ) <-> ( x e. A -> ( x e. C /\ -. x e. B ) ) ) )
3		eldif 2991	. . . . . . 7 |- ( x e. ( C \ B ) <-> ( x e. C /\ -. x e. B ) )
4	3	imbi2i 224	. . . . . 6 |- ( ( x e. A -> x e. ( C \ B ) ) <-> ( x e. A -> ( x e. C /\ -. x e. B ) ) )
5	2, 4	syl6bbr 196	. . . . 5 |- ( ( x e. A -> x e. C ) -> ( ( x e. A -> -. x e. B ) <-> ( x e. A -> x e. ( C \ B ) ) ) )
6	5	sps 1471	. . . 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 119	. . 3 |- ( A C_ C -> ( ( x e. A -> -. x e. B ) <-> ( x e. A -> x e. ( C \ B ) ) ) )
8	7	albidv 1747	. 2 |- ( A C_ C -> ( A. x ( x e. A -> -. x e. B ) <-> A. x ( x e. A -> x e. ( C \ B ) ) ) )
9		disj1 3310	. 2 |- ( ( A i^i B ) = (/) <-> A. x ( x e. A -> -. x e. B ) )
10		dfss2 2997	. 2 |- ( A C_ ( C \ B ) <-> A. x ( x e. A -> x e. ( C \ B ) ) )
11	8, 9, 10	3bitr4g 221	1 |- ( A C_ C -> ( ( A i^i B ) = (/) <-> A C_ ( C \ B ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 102   <-> wb 103   A.wal 1283   = wceq 1285   e. wcel 1434   \ cdif 2979   i^i cin 2981   C_ wss 2982   (/)c0 3267

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065

This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-v 2612  df-dif 2984  df-in 2988  df-ss 2995  df-nul 3268

This theorem is referenced by:  disj2  3315  ssdifsn  3536