Theorem disjpr2 4248

Description: Two completely distinct unordered pairs are disjoint. (Contributed by Alexander van der Vekens, 11-Nov-2017.) (Proof shortened by JJ, 23-Jul-2021.)

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 4180	. . . 4 |- { C , D } = ( { C } u. { D } )
2	1	ineq2i 3811	. . 3 |- ( { A , B } i^i { C , D } ) = ( { A , B } i^i ( { C } u. { D } ) )
3		indi 3873	. . 3 |- ( { A , B } i^i ( { C } u. { D } ) ) = ( ( { A , B } i^i { C } ) u. ( { A , B } i^i { D } ) )
4	2, 3	eqtri 2644	. 2 |- ( { A , B } i^i { C , D } ) = ( ( { A , B } i^i { C } ) u. ( { A , B } i^i { D } ) )
5		df-pr 4180	. . . . . . . 8 |- { A , B } = ( { A } u. { B } )
6	5	ineq1i 3810	. . . . . . 7 |- ( { A , B } i^i { C } ) = ( ( { A } u. { B } ) i^i { C } )
7		indir 3875	. . . . . . 7 |- ( ( { A } u. { B } ) i^i { C } ) = ( ( { A } i^i { C } ) u. ( { B } i^i { C } ) )
8	6, 7	eqtri 2644	. . . . . 6 |- ( { A , B } i^i { C } ) = ( ( { A } i^i { C } ) u. ( { B } i^i { C } ) )
9		disjsn2 4247	. . . . . . . 8 |- ( A =/= C -> ( { A } i^i { C } ) = (/) )
10		disjsn2 4247	. . . . . . . 8 |- ( B =/= C -> ( { B } i^i { C } ) = (/) )
11	9, 10	anim12i 590	. . . . . . 7 |- ( ( A =/= C /\ B =/= C ) -> ( ( { A } i^i { C } ) = (/) /\ ( { B } i^i { C } ) = (/) ) )
12		un00 4011	. . . . . . 7 |- ( ( ( { A } i^i { C } ) = (/) /\ ( { B } i^i { C } ) = (/) ) <-> ( ( { A } i^i { C } ) u. ( { B } i^i { C } ) ) = (/) )
13	11, 12	sylib 208	. . . . . 6 |- ( ( A =/= C /\ B =/= C ) -> ( ( { A } i^i { C } ) u. ( { B } i^i { C } ) ) = (/) )
14	8, 13	syl5eq 2668	. . . . 5 |- ( ( A =/= C /\ B =/= C ) -> ( { A , B } i^i { C } ) = (/) )
15	14	adantr 481	. . . 4 |- ( ( ( A =/= C /\ B =/= C ) /\ ( A =/= D /\ B =/= D ) ) -> ( { A , B } i^i { C } ) = (/) )
16	5	ineq1i 3810	. . . . . . 7 |- ( { A , B } i^i { D } ) = ( ( { A } u. { B } ) i^i { D } )
17		indir 3875	. . . . . . 7 |- ( ( { A } u. { B } ) i^i { D } ) = ( ( { A } i^i { D } ) u. ( { B } i^i { D } ) )
18	16, 17	eqtri 2644	. . . . . 6 |- ( { A , B } i^i { D } ) = ( ( { A } i^i { D } ) u. ( { B } i^i { D } ) )
19		disjsn2 4247	. . . . . . . 8 |- ( A =/= D -> ( { A } i^i { D } ) = (/) )
20		disjsn2 4247	. . . . . . . 8 |- ( B =/= D -> ( { B } i^i { D } ) = (/) )
21	19, 20	anim12i 590	. . . . . . 7 |- ( ( A =/= D /\ B =/= D ) -> ( ( { A } i^i { D } ) = (/) /\ ( { B } i^i { D } ) = (/) ) )
22		un00 4011	. . . . . . 7 |- ( ( ( { A } i^i { D } ) = (/) /\ ( { B } i^i { D } ) = (/) ) <-> ( ( { A } i^i { D } ) u. ( { B } i^i { D } ) ) = (/) )
23	21, 22	sylib 208	. . . . . 6 |- ( ( A =/= D /\ B =/= D ) -> ( ( { A } i^i { D } ) u. ( { B } i^i { D } ) ) = (/) )
24	18, 23	syl5eq 2668	. . . . 5 |- ( ( A =/= D /\ B =/= D ) -> ( { A , B } i^i { D } ) = (/) )
25	24	adantl 482	. . . 4 |- ( ( ( A =/= C /\ B =/= C ) /\ ( A =/= D /\ B =/= D ) ) -> ( { A , B } i^i { D } ) = (/) )
26	15, 25	uneq12d 3768	. . 3 |- ( ( ( A =/= C /\ B =/= C ) /\ ( A =/= D /\ B =/= D ) ) -> ( ( { A , B } i^i { C } ) u. ( { A , B } i^i { D } ) ) = ( (/) u. (/) ) )
27		un0 3967	. . 3 |- ( (/) u. (/) ) = (/)
28	26, 27	syl6eq 2672	. 2 |- ( ( ( A =/= C /\ B =/= C ) /\ ( A =/= D /\ B =/= D ) ) -> ( ( { A , B } i^i { C } ) u. ( { A , B } i^i { D } ) ) = (/) )
29	4, 28	syl5eq 2668	1 |- ( ( ( A =/= C /\ B =/= C ) /\ ( A =/= D /\ B =/= D ) ) -> ( { A , B } i^i { C , D } ) = (/) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   =/= wne 2794   u. cun 3572   i^i cin 3573   (/)c0 3915   {csn 4177   {cpr 4179

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

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-sn 4178  df-pr 4180

This theorem is referenced by:  disjprsn  4250  disjtp2  4252  funcnvqp  5952  funcnvqpOLD  5953