Theorem disjpr2 3657

Description: The intersection of distinct unordered pairs is disjoint. (Contributed by Alexander van der Vekens, 11-Nov-2017.)

Assertion

Ref	Expression
disjpr2	|- ( ( ( A =/= C /\ B =/= C ) /\ ( A =/= D /\ B =/= D ) ) -> ( { A , B } i^i { C , D } ) = (/) )

Proof of Theorem disjpr2

Step	Hyp	Ref	Expression
1		df-pr 3600	. . . 4 |- { C , D } = ( { C } u. { D } )
2	1	a1i 9	. . 3 |- ( ( ( A =/= C /\ B =/= C ) /\ ( A =/= D /\ B =/= D ) ) -> { C , D } = ( { C } u. { D } ) )
3	2	ineq2d 3337	. 2 |- ( ( ( A =/= C /\ B =/= C ) /\ ( A =/= D /\ B =/= D ) ) -> ( { A , B } i^i { C , D } ) = ( { A , B } i^i ( { C } u. { D } ) ) )
4		indi 3383	. . 3 |- ( { A , B } i^i ( { C } u. { D } ) ) = ( ( { A , B } i^i { C } ) u. ( { A , B } i^i { D } ) )
5		df-pr 3600	. . . . . . . 8 |- { A , B } = ( { A } u. { B } )
6	5	ineq1i 3333	. . . . . . 7 |- ( { A , B } i^i { C } ) = ( ( { A } u. { B } ) i^i { C } )
7		indir 3385	. . . . . . 7 |- ( ( { A } u. { B } ) i^i { C } ) = ( ( { A } i^i { C } ) u. ( { B } i^i { C } ) )
8	6, 7	eqtri 2198	. . . . . 6 |- ( { A , B } i^i { C } ) = ( ( { A } i^i { C } ) u. ( { B } i^i { C } ) )
9		disjsn2 3656	. . . . . . . . . 10 |- ( A =/= C -> ( { A } i^i { C } ) = (/) )
10	9	adantr 276	. . . . . . . . 9 |- ( ( A =/= C /\ B =/= C ) -> ( { A } i^i { C } ) = (/) )
11	10	adantr 276	. . . . . . . 8 |- ( ( ( A =/= C /\ B =/= C ) /\ ( A =/= D /\ B =/= D ) ) -> ( { A } i^i { C } ) = (/) )
12		disjsn2 3656	. . . . . . . . . 10 |- ( B =/= C -> ( { B } i^i { C } ) = (/) )
13	12	adantl 277	. . . . . . . . 9 |- ( ( A =/= C /\ B =/= C ) -> ( { B } i^i { C } ) = (/) )
14	13	adantr 276	. . . . . . . 8 |- ( ( ( A =/= C /\ B =/= C ) /\ ( A =/= D /\ B =/= D ) ) -> ( { B } i^i { C } ) = (/) )
15	11, 14	jca 306	. . . . . . 7 |- ( ( ( A =/= C /\ B =/= C ) /\ ( A =/= D /\ B =/= D ) ) -> ( ( { A } i^i { C } ) = (/) /\ ( { B } i^i { C } ) = (/) ) )
16		un00 3470	. . . . . . 7 |- ( ( ( { A } i^i { C } ) = (/) /\ ( { B } i^i { C } ) = (/) ) <-> ( ( { A } i^i { C } ) u. ( { B } i^i { C } ) ) = (/) )
17	15, 16	sylib 122	. . . . . 6 |- ( ( ( A =/= C /\ B =/= C ) /\ ( A =/= D /\ B =/= D ) ) -> ( ( { A } i^i { C } ) u. ( { B } i^i { C } ) ) = (/) )
18	8, 17	eqtrid 2222	. . . . 5 |- ( ( ( A =/= C /\ B =/= C ) /\ ( A =/= D /\ B =/= D ) ) -> ( { A , B } i^i { C } ) = (/) )
19	5	ineq1i 3333	. . . . . . 7 |- ( { A , B } i^i { D } ) = ( ( { A } u. { B } ) i^i { D } )
20		indir 3385	. . . . . . 7 |- ( ( { A } u. { B } ) i^i { D } ) = ( ( { A } i^i { D } ) u. ( { B } i^i { D } ) )
21	19, 20	eqtri 2198	. . . . . 6 |- ( { A , B } i^i { D } ) = ( ( { A } i^i { D } ) u. ( { B } i^i { D } ) )
22		disjsn2 3656	. . . . . . . . . 10 |- ( A =/= D -> ( { A } i^i { D } ) = (/) )
23	22	adantr 276	. . . . . . . . 9 |- ( ( A =/= D /\ B =/= D ) -> ( { A } i^i { D } ) = (/) )
24	23	adantl 277	. . . . . . . 8 |- ( ( ( A =/= C /\ B =/= C ) /\ ( A =/= D /\ B =/= D ) ) -> ( { A } i^i { D } ) = (/) )
25		disjsn2 3656	. . . . . . . . . 10 |- ( B =/= D -> ( { B } i^i { D } ) = (/) )
26	25	adantl 277	. . . . . . . . 9 |- ( ( A =/= D /\ B =/= D ) -> ( { B } i^i { D } ) = (/) )
27	26	adantl 277	. . . . . . . 8 |- ( ( ( A =/= C /\ B =/= C ) /\ ( A =/= D /\ B =/= D ) ) -> ( { B } i^i { D } ) = (/) )
28	24, 27	jca 306	. . . . . . 7 |- ( ( ( A =/= C /\ B =/= C ) /\ ( A =/= D /\ B =/= D ) ) -> ( ( { A } i^i { D } ) = (/) /\ ( { B } i^i { D } ) = (/) ) )
29		un00 3470	. . . . . . 7 |- ( ( ( { A } i^i { D } ) = (/) /\ ( { B } i^i { D } ) = (/) ) <-> ( ( { A } i^i { D } ) u. ( { B } i^i { D } ) ) = (/) )
30	28, 29	sylib 122	. . . . . 6 |- ( ( ( A =/= C /\ B =/= C ) /\ ( A =/= D /\ B =/= D ) ) -> ( ( { A } i^i { D } ) u. ( { B } i^i { D } ) ) = (/) )
31	21, 30	eqtrid 2222	. . . . 5 |- ( ( ( A =/= C /\ B =/= C ) /\ ( A =/= D /\ B =/= D ) ) -> ( { A , B } i^i { D } ) = (/) )
32	18, 31	uneq12d 3291	. . . 4 |- ( ( ( A =/= C /\ B =/= C ) /\ ( A =/= D /\ B =/= D ) ) -> ( ( { A , B } i^i { C } ) u. ( { A , B } i^i { D } ) ) = ( (/) u. (/) ) )
33		un0 3457	. . . 4 |- ( (/) u. (/) ) = (/)
34	32, 33	eqtrdi 2226	. . 3 |- ( ( ( A =/= C /\ B =/= C ) /\ ( A =/= D /\ B =/= D ) ) -> ( ( { A , B } i^i { C } ) u. ( { A , B } i^i { D } ) ) = (/) )
35	4, 34	eqtrid 2222	. 2 |- ( ( ( A =/= C /\ B =/= C ) /\ ( A =/= D /\ B =/= D ) ) -> ( { A , B } i^i ( { C } u. { D } ) ) = (/) )
36	3, 35	eqtrd 2210	1 |- ( ( ( A =/= C /\ B =/= C ) /\ ( A =/= D /\ B =/= D ) ) -> ( { A , B } i^i { C , D } ) = (/) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1353   =/= wne 2347   u. cun 3128   i^i cin 3129   (/)c0 3423   {csn 3593   {cpr 3594

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-v 2740  df-dif 3132  df-un 3134  df-in 3136  df-ss 3143  df-nul 3424  df-sn 3599  df-pr 3600

This theorem is referenced by: (None)