Theorem difex2 6969

Description: If the subtrahend of a class difference exists, then the minuend exists iff the difference exists. (Contributed by NM, 12-Nov-2003.) (Proof shortened by Andrew Salmon, 12-Aug-2011.)

Assertion

Ref	Expression
difex2	|- ( B e. C -> ( A e. _V <-> ( A \ B ) e. _V ) )

Proof of Theorem difex2

Step	Hyp	Ref	Expression
1		difexg 4808	. 2 |- ( A e. _V -> ( A \ B ) e. _V )
2		ssun2 3777	. . . . 5 |- A C_ ( B u. A )
3		uncom 3757	. . . . . 6 |- ( ( A \ B ) u. B ) = ( B u. ( A \ B ) )
4		undif2 4044	. . . . . 6 |- ( B u. ( A \ B ) ) = ( B u. A )
5	3, 4	eqtr2i 2645	. . . . 5 |- ( B u. A ) = ( ( A \ B ) u. B )
6	2, 5	sseqtri 3637	. . . 4 |- A C_ ( ( A \ B ) u. B )
7		unexg 6959	. . . 4 |- ( ( ( A \ B ) e. _V /\ B e. C ) -> ( ( A \ B ) u. B ) e. _V )
8		ssexg 4804	. . . 4 |- ( ( A C_ ( ( A \ B ) u. B ) /\ ( ( A \ B ) u. B ) e. _V ) -> A e. _V )
9	6, 7, 8	sylancr 695	. . 3 |- ( ( ( A \ B ) e. _V /\ B e. C ) -> A e. _V )
10	9	expcom 451	. 2 |- ( B e. C -> ( ( A \ B ) e. _V -> A e. _V ) )
11	1, 10	impbid2 216	1 |- ( B e. C -> ( A e. _V <-> ( A \ B ) e. _V ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   e. wcel 1990   _Vcvv 3200   \ cdif 3571   u. cun 3572   C_ wss 3574

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-sn 4178  df-pr 4180  df-uni 4437

This theorem is referenced by:  elpwun  6977