Theorem difin2 3890

Description: Represent a class difference as an intersection with a larger difference. (Contributed by Jeff Madsen, 2-Sep-2009.)

Assertion

Ref	Expression
difin2	|- ( A C_ C -> ( A \ B ) = ( ( C \ B ) i^i A ) )

Proof of Theorem difin2

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ssel 3597	. . . . 5 |- ( A C_ C -> ( x e. A -> x e. C ) )
2	1	pm4.71d 666	. . . 4 |- ( A C_ C -> ( x e. A <-> ( x e. A /\ x e. C ) ) )
3	2	anbi1d 741	. . 3 |- ( A C_ C -> ( ( x e. A /\ -. x e. B ) <-> ( ( x e. A /\ x e. C ) /\ -. x e. B ) ) )
4		eldif 3584	. . 3 |- ( x e. ( A \ B ) <-> ( x e. A /\ -. x e. B ) )
5		elin 3796	. . . 4 |- ( x e. ( ( C \ B ) i^i A ) <-> ( x e. ( C \ B ) /\ x e. A ) )
6		eldif 3584	. . . . 5 |- ( x e. ( C \ B ) <-> ( x e. C /\ -. x e. B ) )
7	6	anbi1i 731	. . . 4 |- ( ( x e. ( C \ B ) /\ x e. A ) <-> ( ( x e. C /\ -. x e. B ) /\ x e. A ) )
8		ancom 466	. . . . 5 |- ( ( ( x e. C /\ -. x e. B ) /\ x e. A ) <-> ( x e. A /\ ( x e. C /\ -. x e. B ) ) )
9		anass 681	. . . . 5 |- ( ( ( x e. A /\ x e. C ) /\ -. x e. B ) <-> ( x e. A /\ ( x e. C /\ -. x e. B ) ) )
10	8, 9	bitr4i 267	. . . 4 |- ( ( ( x e. C /\ -. x e. B ) /\ x e. A ) <-> ( ( x e. A /\ x e. C ) /\ -. x e. B ) )
11	5, 7, 10	3bitri 286	. . 3 |- ( x e. ( ( C \ B ) i^i A ) <-> ( ( x e. A /\ x e. C ) /\ -. x e. B ) )
12	3, 4, 11	3bitr4g 303	. 2 |- ( A C_ C -> ( x e. ( A \ B ) <-> x e. ( ( C \ B ) i^i A ) ) )
13	12	eqrdv 2620	1 |- ( A C_ C -> ( A \ B ) = ( ( C \ B ) i^i A ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   \ cdif 3571   i^i cin 3573   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-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

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-v 3202  df-dif 3577  df-in 3581  df-ss 3588

This theorem is referenced by:  gsumdifsnd  18360  issubdrg  18805  restcld  20976  limcnlp  23642  difelsiga  30196  sigapildsyslem  30224  ldgenpisyslem1  30226  difelcarsg2  30375  ballotlemfp1  30553  asindmre  33495  caragendifcl  40728  gsumdifsndf  42144