Theorem disjtp2 4252

Description: Two completely distinct unordered triples are disjoint. (Contributed by AV, 14-Nov-2021.)

Assertion

Ref	Expression
disjtp2	|- ( ( ( A =/= D /\ B =/= D /\ C =/= D ) /\ ( A =/= E /\ B =/= E /\ C =/= E ) /\ ( A =/= F /\ B =/= F /\ C =/= F ) ) -> ( { A , B , C } i^i { D , E , F } ) = (/) )

Proof of Theorem disjtp2

Step	Hyp	Ref	Expression
1		df-tp 4182	. . 3 |- { D , E , F } = ( { D , E } u. { F } )
2	1	ineq2i 3811	. 2 |- ( { A , B , C } i^i { D , E , F } ) = ( { A , B , C } i^i ( { D , E } u. { F } ) )
3		df-tp 4182	. . . . . 6 |- { A , B , C } = ( { A , B } u. { C } )
4	3	ineq1i 3810	. . . . 5 |- ( { A , B , C } i^i { D , E } ) = ( ( { A , B } u. { C } ) i^i { D , E } )
5		3simpa 1058	. . . . . . . . 9 |- ( ( A =/= D /\ B =/= D /\ C =/= D ) -> ( A =/= D /\ B =/= D ) )
6		3simpa 1058	. . . . . . . . 9 |- ( ( A =/= E /\ B =/= E /\ C =/= E ) -> ( A =/= E /\ B =/= E ) )
7		disjpr2 4248	. . . . . . . . 9 |- ( ( ( A =/= D /\ B =/= D ) /\ ( A =/= E /\ B =/= E ) ) -> ( { A , B } i^i { D , E } ) = (/) )
8	5, 6, 7	syl2an 494	. . . . . . . 8 |- ( ( ( A =/= D /\ B =/= D /\ C =/= D ) /\ ( A =/= E /\ B =/= E /\ C =/= E ) ) -> ( { A , B } i^i { D , E } ) = (/) )
9	8	3adant3 1081	. . . . . . 7 |- ( ( ( A =/= D /\ B =/= D /\ C =/= D ) /\ ( A =/= E /\ B =/= E /\ C =/= E ) /\ ( A =/= F /\ B =/= F /\ C =/= F ) ) -> ( { A , B } i^i { D , E } ) = (/) )
10		incom 3805	. . . . . . . 8 |- ( { C } i^i { D , E } ) = ( { D , E } i^i { C } )
11		necom 2847	. . . . . . . . . . . 12 |- ( C =/= D <-> D =/= C )
12	11	biimpi 206	. . . . . . . . . . 11 |- ( C =/= D -> D =/= C )
13	12	3ad2ant3 1084	. . . . . . . . . 10 |- ( ( A =/= D /\ B =/= D /\ C =/= D ) -> D =/= C )
14		necom 2847	. . . . . . . . . . . 12 |- ( C =/= E <-> E =/= C )
15	14	biimpi 206	. . . . . . . . . . 11 |- ( C =/= E -> E =/= C )
16	15	3ad2ant3 1084	. . . . . . . . . 10 |- ( ( A =/= E /\ B =/= E /\ C =/= E ) -> E =/= C )
17		disjprsn 4250	. . . . . . . . . 10 |- ( ( D =/= C /\ E =/= C ) -> ( { D , E } i^i { C } ) = (/) )
18	13, 16, 17	syl2an 494	. . . . . . . . 9 |- ( ( ( A =/= D /\ B =/= D /\ C =/= D ) /\ ( A =/= E /\ B =/= E /\ C =/= E ) ) -> ( { D , E } i^i { C } ) = (/) )
19	18	3adant3 1081	. . . . . . . 8 |- ( ( ( A =/= D /\ B =/= D /\ C =/= D ) /\ ( A =/= E /\ B =/= E /\ C =/= E ) /\ ( A =/= F /\ B =/= F /\ C =/= F ) ) -> ( { D , E } i^i { C } ) = (/) )
20	10, 19	syl5eq 2668	. . . . . . 7 |- ( ( ( A =/= D /\ B =/= D /\ C =/= D ) /\ ( A =/= E /\ B =/= E /\ C =/= E ) /\ ( A =/= F /\ B =/= F /\ C =/= F ) ) -> ( { C } i^i { D , E } ) = (/) )
21	9, 20	jca 554	. . . . . 6 |- ( ( ( A =/= D /\ B =/= D /\ C =/= D ) /\ ( A =/= E /\ B =/= E /\ C =/= E ) /\ ( A =/= F /\ B =/= F /\ C =/= F ) ) -> ( ( { A , B } i^i { D , E } ) = (/) /\ ( { C } i^i { D , E } ) = (/) ) )
22		undisj1 4029	. . . . . 6 |- ( ( ( { A , B } i^i { D , E } ) = (/) /\ ( { C } i^i { D , E } ) = (/) ) <-> ( ( { A , B } u. { C } ) i^i { D , E } ) = (/) )
23	21, 22	sylib 208	. . . . 5 |- ( ( ( A =/= D /\ B =/= D /\ C =/= D ) /\ ( A =/= E /\ B =/= E /\ C =/= E ) /\ ( A =/= F /\ B =/= F /\ C =/= F ) ) -> ( ( { A , B } u. { C } ) i^i { D , E } ) = (/) )
24	4, 23	syl5eq 2668	. . . 4 |- ( ( ( A =/= D /\ B =/= D /\ C =/= D ) /\ ( A =/= E /\ B =/= E /\ C =/= E ) /\ ( A =/= F /\ B =/= F /\ C =/= F ) ) -> ( { A , B , C } i^i { D , E } ) = (/) )
25		disjtpsn 4251	. . . . 5 |- ( ( A =/= F /\ B =/= F /\ C =/= F ) -> ( { A , B , C } i^i { F } ) = (/) )
26	25	3ad2ant3 1084	. . . 4 |- ( ( ( A =/= D /\ B =/= D /\ C =/= D ) /\ ( A =/= E /\ B =/= E /\ C =/= E ) /\ ( A =/= F /\ B =/= F /\ C =/= F ) ) -> ( { A , B , C } i^i { F } ) = (/) )
27	24, 26	jca 554	. . 3 |- ( ( ( A =/= D /\ B =/= D /\ C =/= D ) /\ ( A =/= E /\ B =/= E /\ C =/= E ) /\ ( A =/= F /\ B =/= F /\ C =/= F ) ) -> ( ( { A , B , C } i^i { D , E } ) = (/) /\ ( { A , B , C } i^i { F } ) = (/) ) )
28		undisj2 4030	. . 3 |- ( ( ( { A , B , C } i^i { D , E } ) = (/) /\ ( { A , B , C } i^i { F } ) = (/) ) <-> ( { A , B , C } i^i ( { D , E } u. { F } ) ) = (/) )
29	27, 28	sylib 208	. 2 |- ( ( ( A =/= D /\ B =/= D /\ C =/= D ) /\ ( A =/= E /\ B =/= E /\ C =/= E ) /\ ( A =/= F /\ B =/= F /\ C =/= F ) ) -> ( { A , B , C } i^i ( { D , E } u. { F } ) ) = (/) )
30	2, 29	syl5eq 2668	1 |- ( ( ( A =/= D /\ B =/= D /\ C =/= D ) /\ ( A =/= E /\ B =/= E /\ C =/= E ) /\ ( A =/= F /\ B =/= F /\ C =/= F ) ) -> ( { A , B , C } i^i { D , E , F } ) = (/) )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   =/= wne 2794   u. cun 3572   i^i cin 3573   (/)c0 3915   {csn 4177   {cpr 4179   {ctp 4181

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-3an 1039  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  df-tp 4182

This theorem is referenced by:  cnfldfun  19758