Theorem disjdifprg 29388

Description: A trivial partition into a subset and its complement. (Contributed by Thierry Arnoux, 25-Dec-2016.)

Assertion

Ref	Expression
disjdifprg	|- ( ( A e. V /\ B e. W ) -> Disj x e. { ( B \ A ) , A } x )

Distinct variable groups:   x, A   x, B

Allowed substitution hints:   V( x)   W( x)

Proof of Theorem disjdifprg

Step	Hyp	Ref	Expression
1		disjxsn 4646	. . . . . 6 |- Disj x e. { (/) } x
2		simpr 477	. . . . . . . 8 |- ( ( B e. W /\ B = (/) ) -> B = (/) )
3		eqidd 2623	. . . . . . . 8 |- ( ( B e. W /\ B = (/) ) -> (/) = (/) )
4		elex 3212	. . . . . . . . . 10 |- ( B e. W -> B e. _V )
5		0ex 4790	. . . . . . . . . . 11 |- (/) e. _V
6	5	a1i 11	. . . . . . . . . 10 |- ( B e. W -> (/) e. _V )
7	4, 6, 6	preqsnd 4392	. . . . . . . . 9 |- ( B e. W -> ( { B , (/) } = { (/) } <-> ( B = (/) /\ (/) = (/) ) ) )
8	7	adantr 481	. . . . . . . 8 |- ( ( B e. W /\ B = (/) ) -> ( { B , (/) } = { (/) } <-> ( B = (/) /\ (/) = (/) ) ) )
9	2, 3, 8	mpbir2and 957	. . . . . . 7 |- ( ( B e. W /\ B = (/) ) -> { B , (/) } = { (/) } )
10	9	disjeq1d 4628	. . . . . 6 |- ( ( B e. W /\ B = (/) ) -> (Disj x e. { B , (/) } x <-> Disj x e. { (/) } x ) )
11	1, 10	mpbiri 248	. . . . 5 |- ( ( B e. W /\ B = (/) ) -> Disj x e. { B , (/) } x )
12		in0 3968	. . . . . 6 |- ( B i^i (/) ) = (/)
13	4	adantr 481	. . . . . . 7 |- ( ( B e. W /\ B =/= (/) ) -> B e. _V )
14	5	a1i 11	. . . . . . 7 |- ( ( B e. W /\ B =/= (/) ) -> (/) e. _V )
15		simpr 477	. . . . . . 7 |- ( ( B e. W /\ B =/= (/) ) -> B =/= (/) )
16		id 22	. . . . . . . 8 |- ( x = B -> x = B )
17		id 22	. . . . . . . 8 |- ( x = (/) -> x = (/) )
18	16, 17	disjprg 4648	. . . . . . 7 |- ( ( B e. _V /\ (/) e. _V /\ B =/= (/) ) -> (Disj x e. { B , (/) } x <-> ( B i^i (/) ) = (/) ) )
19	13, 14, 15, 18	syl3anc 1326	. . . . . 6 |- ( ( B e. W /\ B =/= (/) ) -> (Disj x e. { B , (/) } x <-> ( B i^i (/) ) = (/) ) )
20	12, 19	mpbiri 248	. . . . 5 |- ( ( B e. W /\ B =/= (/) ) -> Disj x e. { B , (/) } x )
21	11, 20	pm2.61dane 2881	. . . 4 |- ( B e. W -> Disj x e. { B , (/) } x )
22	21	ad2antlr 763	. . 3 |- ( ( ( A e. V /\ B e. W ) /\ A = (/) ) -> Disj x e. { B , (/) } x )
23		difeq2 3722	. . . . . . 7 |- ( A = (/) -> ( B \ A ) = ( B \ (/) ) )
24		dif0 3950	. . . . . . 7 |- ( B \ (/) ) = B
25	23, 24	syl6eq 2672	. . . . . 6 |- ( A = (/) -> ( B \ A ) = B )
26		id 22	. . . . . 6 |- ( A = (/) -> A = (/) )
27	25, 26	preq12d 4276	. . . . 5 |- ( A = (/) -> { ( B \ A ) , A } = { B , (/) } )
28	27	disjeq1d 4628	. . . 4 |- ( A = (/) -> (Disj x e. { ( B \ A ) , A } x <-> Disj x e. { B , (/) } x ) )
29	28	adantl 482	. . 3 |- ( ( ( A e. V /\ B e. W ) /\ A = (/) ) -> (Disj x e. { ( B \ A ) , A } x <-> Disj x e. { B , (/) } x ) )
30	22, 29	mpbird 247	. 2 |- ( ( ( A e. V /\ B e. W ) /\ A = (/) ) -> Disj x e. { ( B \ A ) , A } x )
31		incom 3805	. . . 4 |- ( A i^i ( B \ A ) ) = ( ( B \ A ) i^i A )
32		disjdif 4040	. . . 4 |- ( A i^i ( B \ A ) ) = (/)
33	31, 32	eqtr3i 2646	. . 3 |- ( ( B \ A ) i^i A ) = (/)
34		difexg 4808	. . . . 5 |- ( B e. W -> ( B \ A ) e. _V )
35	34	ad2antlr 763	. . . 4 |- ( ( ( A e. V /\ B e. W ) /\ -. A = (/) ) -> ( B \ A ) e. _V )
36		elex 3212	. . . . 5 |- ( A e. V -> A e. _V )
37	36	ad2antrr 762	. . . 4 |- ( ( ( A e. V /\ B e. W ) /\ -. A = (/) ) -> A e. _V )
38		ssid 3624	. . . . . 6 |- ( B \ A ) C_ ( B \ A )
39		ssdifeq0 4051	. . . . . . . 8 |- ( A C_ ( B \ A ) <-> A = (/) )
40	39	notbii 310	. . . . . . 7 |- ( -. A C_ ( B \ A ) <-> -. A = (/) )
41		nssne2 3662	. . . . . . 7 |- ( ( ( B \ A ) C_ ( B \ A ) /\ -. A C_ ( B \ A ) ) -> ( B \ A ) =/= A )
42	40, 41	sylan2br 493	. . . . . 6 |- ( ( ( B \ A ) C_ ( B \ A ) /\ -. A = (/) ) -> ( B \ A ) =/= A )
43	38, 42	mpan 706	. . . . 5 |- ( -. A = (/) -> ( B \ A ) =/= A )
44	43	adantl 482	. . . 4 |- ( ( ( A e. V /\ B e. W ) /\ -. A = (/) ) -> ( B \ A ) =/= A )
45		id 22	. . . . 5 |- ( x = ( B \ A ) -> x = ( B \ A ) )
46		id 22	. . . . 5 |- ( x = A -> x = A )
47	45, 46	disjprg 4648	. . . 4 |- ( ( ( B \ A ) e. _V /\ A e. _V /\ ( B \ A ) =/= A ) -> (Disj x e. { ( B \ A ) , A } x <-> ( ( B \ A ) i^i A ) = (/) ) )
48	35, 37, 44, 47	syl3anc 1326	. . 3 |- ( ( ( A e. V /\ B e. W ) /\ -. A = (/) ) -> (Disj x e. { ( B \ A ) , A } x <-> ( ( B \ A ) i^i A ) = (/) ) )
49	33, 48	mpbiri 248	. 2 |- ( ( ( A e. V /\ B e. W ) /\ -. A = (/) ) -> Disj x e. { ( B \ A ) , A } x )
50	30, 49	pm2.61dan 832	1 |- ( ( A e. V /\ B e. W ) -> Disj x e. { ( B \ A ) , A } x )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   _Vcvv 3200   \ cdif 3571   i^i cin 3573   C_ wss 3574   (/)c0 3915   {csn 4177   {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:  disjdifprg2  29389  measssd  30278