Theorem disjpr2 3434

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 3382	. . . 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 3138	. 2 |- ( ( ( A =/= C /\ B =/= C ) /\ ( A =/= D /\ B =/= D ) ) -> ( { A , B } i^i { C , D } ) = ( { A , B } i^i ( { C } u. { D } ) ) )
4		indi 3184	. . 3 |- ( { A , B } i^i ( { C } u. { D } ) ) = ( ( { A , B } i^i { C } ) u. ( { A , B } i^i { D } ) )
5		df-pr 3382	. . . . . . . 8 |- { A , B } = ( { A } u. { B } )
6	5	ineq1i 3134	. . . . . . 7 |- ( { A , B } i^i { C } ) = ( ( { A } u. { B } ) i^i { C } )
7		indir 3186	. . . . . . 7 |- ( ( { A } u. { B } ) i^i { C } ) = ( ( { A } i^i { C } ) u. ( { B } i^i { C } ) )
8	6, 7	eqtri 2060	. . . . . 6 |- ( { A , B } i^i { C } ) = ( ( { A } i^i { C } ) u. ( { B } i^i { C } ) )
9		disjsn2 3433	. . . . . . . . . 10 |- ( A =/= C -> ( { A } i^i { C } ) = (/) )
10	9	adantr 261	. . . . . . . . 9 |- ( ( A =/= C /\ B =/= C ) -> ( { A } i^i { C } ) = (/) )
11	10	adantr 261	. . . . . . . 8 |- ( ( ( A =/= C /\ B =/= C ) /\ ( A =/= D /\ B =/= D ) ) -> ( { A } i^i { C } ) = (/) )
12		disjsn2 3433	. . . . . . . . . 10 |- ( B =/= C -> ( { B } i^i { C } ) = (/) )
13	12	adantl 262	. . . . . . . . 9 |- ( ( A =/= C /\ B =/= C ) -> ( { B } i^i { C } ) = (/) )
14	13	adantr 261	. . . . . . . 8 |- ( ( ( A =/= C /\ B =/= C ) /\ ( A =/= D /\ B =/= D ) ) -> ( { B } i^i { C } ) = (/) )
15	11, 14	jca 290	. . . . . . 7 |- ( ( ( A =/= C /\ B =/= C ) /\ ( A =/= D /\ B =/= D ) ) -> ( ( { A } i^i { C } ) = (/) /\ ( { B } i^i { C } ) = (/) ) )
16		un00 3263	. . . . . . 7 |- ( ( ( { A } i^i { C } ) = (/) /\ ( { B } i^i { C } ) = (/) ) <-> ( ( { A } i^i { C } ) u. ( { B } i^i { C } ) ) = (/) )
17	15, 16	sylib 127	. . . . . 6 |- ( ( ( A =/= C /\ B =/= C ) /\ ( A =/= D /\ B =/= D ) ) -> ( ( { A } i^i { C } ) u. ( { B } i^i { C } ) ) = (/) )
18	8, 17	syl5eq 2084	. . . . 5 |- ( ( ( A =/= C /\ B =/= C ) /\ ( A =/= D /\ B =/= D ) ) -> ( { A , B } i^i { C } ) = (/) )
19	5	ineq1i 3134	. . . . . . 7 |- ( { A , B } i^i { D } ) = ( ( { A } u. { B } ) i^i { D } )
20		indir 3186	. . . . . . 7 |- ( ( { A } u. { B } ) i^i { D } ) = ( ( { A } i^i { D } ) u. ( { B } i^i { D } ) )
21	19, 20	eqtri 2060	. . . . . 6 |- ( { A , B } i^i { D } ) = ( ( { A } i^i { D } ) u. ( { B } i^i { D } ) )
22		disjsn2 3433	. . . . . . . . . 10 |- ( A =/= D -> ( { A } i^i { D } ) = (/) )
23	22	adantr 261	. . . . . . . . 9 |- ( ( A =/= D /\ B =/= D ) -> ( { A } i^i { D } ) = (/) )
24	23	adantl 262	. . . . . . . 8 |- ( ( ( A =/= C /\ B =/= C ) /\ ( A =/= D /\ B =/= D ) ) -> ( { A } i^i { D } ) = (/) )
25		disjsn2 3433	. . . . . . . . . 10 |- ( B =/= D -> ( { B } i^i { D } ) = (/) )
26	25	adantl 262	. . . . . . . . 9 |- ( ( A =/= D /\ B =/= D ) -> ( { B } i^i { D } ) = (/) )
27	26	adantl 262	. . . . . . . 8 |- ( ( ( A =/= C /\ B =/= C ) /\ ( A =/= D /\ B =/= D ) ) -> ( { B } i^i { D } ) = (/) )
28	24, 27	jca 290	. . . . . . 7 |- ( ( ( A =/= C /\ B =/= C ) /\ ( A =/= D /\ B =/= D ) ) -> ( ( { A } i^i { D } ) = (/) /\ ( { B } i^i { D } ) = (/) ) )
29		un00 3263	. . . . . . 7 |- ( ( ( { A } i^i { D } ) = (/) /\ ( { B } i^i { D } ) = (/) ) <-> ( ( { A } i^i { D } ) u. ( { B } i^i { D } ) ) = (/) )
30	28, 29	sylib 127	. . . . . 6 |- ( ( ( A =/= C /\ B =/= C ) /\ ( A =/= D /\ B =/= D ) ) -> ( ( { A } i^i { D } ) u. ( { B } i^i { D } ) ) = (/) )
31	21, 30	syl5eq 2084	. . . . 5 |- ( ( ( A =/= C /\ B =/= C ) /\ ( A =/= D /\ B =/= D ) ) -> ( { A , B } i^i { D } ) = (/) )
32	18, 31	uneq12d 3098	. . . 4 |- ( ( ( A =/= C /\ B =/= C ) /\ ( A =/= D /\ B =/= D ) ) -> ( ( { A , B } i^i { C } ) u. ( { A , B } i^i { D } ) ) = ( (/) u. (/) ) )
33		un0 3251	. . . 4 |- ( (/) u. (/) ) = (/)
34	32, 33	syl6eq 2088	. . 3 |- ( ( ( A =/= C /\ B =/= C ) /\ ( A =/= D /\ B =/= D ) ) -> ( ( { A , B } i^i { C } ) u. ( { A , B } i^i { D } ) ) = (/) )
35	4, 34	syl5eq 2084	. 2 |- ( ( ( A =/= C /\ B =/= C ) /\ ( A =/= D /\ B =/= D ) ) -> ( { A , B } i^i ( { C } u. { D } ) ) = (/) )
36	3, 35	eqtrd 2072	1 |- ( ( ( A =/= C /\ B =/= C ) /\ ( A =/= D /\ B =/= D ) ) -> ( { A , B } i^i { C , D } ) = (/) )

Syntax hints:   -> wi 4   /\ wa 97   = wceq 1243   =/= wne 2204   u. cun 2915   i^i cin 2916   (/)c0 3224   {csn 3375   {cpr 3376

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-ral 2311  df-v 2559  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-sn 3381  df-pr 3382

This theorem is referenced by: (None)