Theorem disjpr2 3623

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 3567	. . . 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 3308	. 2 |- ( ( ( A =/= C /\ B =/= C ) /\ ( A =/= D /\ B =/= D ) ) -> ( { A , B } i^i { C , D } ) = ( { A , B } i^i ( { C } u. { D } ) ) )
4		indi 3354	. . 3 |- ( { A , B } i^i ( { C } u. { D } ) ) = ( ( { A , B } i^i { C } ) u. ( { A , B } i^i { D } ) )
5		df-pr 3567	. . . . . . . 8 |- { A , B } = ( { A } u. { B } )
6	5	ineq1i 3304	. . . . . . 7 |- ( { A , B } i^i { C } ) = ( ( { A } u. { B } ) i^i { C } )
7		indir 3356	. . . . . . 7 |- ( ( { A } u. { B } ) i^i { C } ) = ( ( { A } i^i { C } ) u. ( { B } i^i { C } ) )
8	6, 7	eqtri 2178	. . . . . 6 |- ( { A , B } i^i { C } ) = ( ( { A } i^i { C } ) u. ( { B } i^i { C } ) )
9		disjsn2 3622	. . . . . . . . . 10 |- ( A =/= C -> ( { A } i^i { C } ) = (/) )
10	9	adantr 274	. . . . . . . . 9 |- ( ( A =/= C /\ B =/= C ) -> ( { A } i^i { C } ) = (/) )
11	10	adantr 274	. . . . . . . 8 |- ( ( ( A =/= C /\ B =/= C ) /\ ( A =/= D /\ B =/= D ) ) -> ( { A } i^i { C } ) = (/) )
12		disjsn2 3622	. . . . . . . . . 10 |- ( B =/= C -> ( { B } i^i { C } ) = (/) )
13	12	adantl 275	. . . . . . . . 9 |- ( ( A =/= C /\ B =/= C ) -> ( { B } i^i { C } ) = (/) )
14	13	adantr 274	. . . . . . . 8 |- ( ( ( A =/= C /\ B =/= C ) /\ ( A =/= D /\ B =/= D ) ) -> ( { B } i^i { C } ) = (/) )
15	11, 14	jca 304	. . . . . . 7 |- ( ( ( A =/= C /\ B =/= C ) /\ ( A =/= D /\ B =/= D ) ) -> ( ( { A } i^i { C } ) = (/) /\ ( { B } i^i { C } ) = (/) ) )
16		un00 3440	. . . . . . 7 |- ( ( ( { A } i^i { C } ) = (/) /\ ( { B } i^i { C } ) = (/) ) <-> ( ( { A } i^i { C } ) u. ( { B } i^i { C } ) ) = (/) )
17	15, 16	sylib 121	. . . . . 6 |- ( ( ( A =/= C /\ B =/= C ) /\ ( A =/= D /\ B =/= D ) ) -> ( ( { A } i^i { C } ) u. ( { B } i^i { C } ) ) = (/) )
18	8, 17	syl5eq 2202	. . . . 5 |- ( ( ( A =/= C /\ B =/= C ) /\ ( A =/= D /\ B =/= D ) ) -> ( { A , B } i^i { C } ) = (/) )
19	5	ineq1i 3304	. . . . . . 7 |- ( { A , B } i^i { D } ) = ( ( { A } u. { B } ) i^i { D } )
20		indir 3356	. . . . . . 7 |- ( ( { A } u. { B } ) i^i { D } ) = ( ( { A } i^i { D } ) u. ( { B } i^i { D } ) )
21	19, 20	eqtri 2178	. . . . . 6 |- ( { A , B } i^i { D } ) = ( ( { A } i^i { D } ) u. ( { B } i^i { D } ) )
22		disjsn2 3622	. . . . . . . . . 10 |- ( A =/= D -> ( { A } i^i { D } ) = (/) )
23	22	adantr 274	. . . . . . . . 9 |- ( ( A =/= D /\ B =/= D ) -> ( { A } i^i { D } ) = (/) )
24	23	adantl 275	. . . . . . . 8 |- ( ( ( A =/= C /\ B =/= C ) /\ ( A =/= D /\ B =/= D ) ) -> ( { A } i^i { D } ) = (/) )
25		disjsn2 3622	. . . . . . . . . 10 |- ( B =/= D -> ( { B } i^i { D } ) = (/) )
26	25	adantl 275	. . . . . . . . 9 |- ( ( A =/= D /\ B =/= D ) -> ( { B } i^i { D } ) = (/) )
27	26	adantl 275	. . . . . . . 8 |- ( ( ( A =/= C /\ B =/= C ) /\ ( A =/= D /\ B =/= D ) ) -> ( { B } i^i { D } ) = (/) )
28	24, 27	jca 304	. . . . . . 7 |- ( ( ( A =/= C /\ B =/= C ) /\ ( A =/= D /\ B =/= D ) ) -> ( ( { A } i^i { D } ) = (/) /\ ( { B } i^i { D } ) = (/) ) )
29		un00 3440	. . . . . . 7 |- ( ( ( { A } i^i { D } ) = (/) /\ ( { B } i^i { D } ) = (/) ) <-> ( ( { A } i^i { D } ) u. ( { B } i^i { D } ) ) = (/) )
30	28, 29	sylib 121	. . . . . 6 |- ( ( ( A =/= C /\ B =/= C ) /\ ( A =/= D /\ B =/= D ) ) -> ( ( { A } i^i { D } ) u. ( { B } i^i { D } ) ) = (/) )
31	21, 30	syl5eq 2202	. . . . 5 |- ( ( ( A =/= C /\ B =/= C ) /\ ( A =/= D /\ B =/= D ) ) -> ( { A , B } i^i { D } ) = (/) )
32	18, 31	uneq12d 3262	. . . 4 |- ( ( ( A =/= C /\ B =/= C ) /\ ( A =/= D /\ B =/= D ) ) -> ( ( { A , B } i^i { C } ) u. ( { A , B } i^i { D } ) ) = ( (/) u. (/) ) )
33		un0 3427	. . . 4 |- ( (/) u. (/) ) = (/)
34	32, 33	eqtrdi 2206	. . 3 |- ( ( ( A =/= C /\ B =/= C ) /\ ( A =/= D /\ B =/= D ) ) -> ( ( { A , B } i^i { C } ) u. ( { A , B } i^i { D } ) ) = (/) )
35	4, 34	syl5eq 2202	. 2 |- ( ( ( A =/= C /\ B =/= C ) /\ ( A =/= D /\ B =/= D ) ) -> ( { A , B } i^i ( { C } u. { D } ) ) = (/) )
36	3, 35	eqtrd 2190	1 |- ( ( ( A =/= C /\ B =/= C ) /\ ( A =/= D /\ B =/= D ) ) -> ( { A , B } i^i { C , D } ) = (/) )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1335   =/= wne 2327   u. cun 3100   i^i cin 3101   (/)c0 3394   {csn 3560   {cpr 3561

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139

This theorem depends on definitions:  df-bi 116  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-ral 2440  df-v 2714  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-nul 3395  df-sn 3566  df-pr 3567

This theorem is referenced by: (None)