Theorem disjdifprg2 29389

Description: A trivial partition of a set into its difference and intersection with another set. (Contributed by Thierry Arnoux, 25-Dec-2016.)

Assertion

Ref	Expression
disjdifprg2	|- ( A e. V -> Disj x e. { ( A \ B ) , ( A i^i B ) } x )

Distinct variable groups:   x, A   x, B

Allowed substitution hint:   V( x)

Proof of Theorem disjdifprg2

Step	Hyp	Ref	Expression
1		inex1g 4801	. . 3 |- ( A e. V -> ( A i^i B ) e. _V )
2		elex 3212	. . 3 |- ( A e. V -> A e. _V )
3		disjdifprg 29388	. . 3 |- ( ( ( A i^i B ) e. _V /\ A e. _V ) -> Disj x e. { ( A \ ( A i^i B ) ) , ( A i^i B ) } x )
4	1, 2, 3	syl2anc 693	. 2 |- ( A e. V -> Disj x e. { ( A \ ( A i^i B ) ) , ( A i^i B ) } x )
5		difin 3861	. . . . 5 |- ( A \ ( A i^i B ) ) = ( A \ B )
6	5	preq1i 4271	. . . 4 |- { ( A \ ( A i^i B ) ) , ( A i^i B ) } = { ( A \ B ) , ( A i^i B ) }
7	6	a1i 11	. . 3 |- ( A e. V -> { ( A \ ( A i^i B ) ) , ( A i^i B ) } = { ( A \ B ) , ( A i^i B ) } )
8	7	disjeq1d 4628	. 2 |- ( A e. V -> (Disj x e. { ( A \ ( A i^i B ) ) , ( A i^i B ) } x <-> Disj x e. { ( A \ B ) , ( A i^i B ) } x ) )
9	4, 8	mpbid 222	1 |- ( A e. V -> Disj x e. { ( A \ B ) , ( A i^i B ) } x )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   _Vcvv 3200   \ cdif 3571   i^i cin 3573   {cpr 4179  Disj wdisj 4620

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  ax-sep 4781  ax-nul 4789

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-sn 4178  df-pr 4180  df-disj 4621

This theorem is referenced by:  measxun2  30273